What we finally came to in our last discussion was an idea (from Tom) that knowledge isn't an internal mental state -- rather, it is a kind of relationship between you and the thing you know about. There's a contemporary school of philosophy that believes just that; but it is also true of the ancient position.
As Aristotle explains in De Anima and elsewhere, knowledge comes to be in us via a process that starts when we encounter the unknown thing. First we must perceive the thing through our senses. Either the sense itself or (in cases where more than one sense is involved) our "common sense" will present us with an image of the thing in our minds. This image in our minds is very similar to what Kant was calling our representation, but for Aristotelians it is not knowledge. Knowledge comes after we use our imagination:
To the thinking soul images serve as if they were contents of perception (and when it asserts or denies them to be good or bad it avoids or pursues them).... The faculty of thinking then thinks the forms in the images, and as in the former case what is to be pursued or avoided is marked out for it, so where there is no sensation and it is engaged upon the images it is moved to pursuit or avoidance. E.g.. perceiving by sense that the beacon is fire, it recognizes in virtue of the general faculty of sense that it signifies an enemy, because it sees it moving; but sometimes by means of the images or thoughts which are within the soul, just as if it were seeing, it calculates and deliberates what is to come by reference to what is present; and when it makes a pronouncement, as in the case of sensation it pronounces the object to be pleasant or painful, in this case it avoids or persues and so generally in cases of action.In other words, we take our initial image and use our imagination to add or subtract qualities. In this process, we sort out what it is that makes the thing that kind of thing -- the purpose, or function, which Aristotle calls the 'final cause.' The beacon can be lit or not; and in sorting out the difference we learn what it is that makes it a beacon and not just a fire (i.e., that it is lit only when the enemy is coming; and thus its final cause is to warn us of invasions). A chair can have two legs or three or four; or it can be blue or red. None of these things causes it to stop being a chair. However, if it is too small, or broken, it cannot be a chair (though it might, in the first case, be a toy chair). A bird can be bigger or smaller (and even flightless!), but it serves a purpose (its own purpose, that is: it sustains itself as a bird, and is involved in the production of more birds of that type).
At this point, we have knowledge. In Aristotle's terms, the final cause is normally also the formal cause -- that it, it is the form of the thing. The form of the chair or the bird comes to be in our minds. That is real knowledge, without mistake: we possess the form.
There are a couple of problems with this approach. It will jump out that Aristotle is using at least one and possibly two invisibles of the type that the West has come to fear since Ockham. "Form" isn't visible except when expressed in matter; the form in our minds is visible only as an image in our minds. Likewise, Aristotle puts all this down to the working of the soul. It seems like we could simply say that he's using "soul" where we would use "mind," but that's not right: the soul turns out to be another form. In fact it is our form, the organizing principle that makes us who we are and gives us our purpose (which, for Aristotle, is to seek understanding through rational activity; but you can take the more pedestrian view that our purpose, as with any animal, is merely to sustain ourselves and produce others like us).
The other problem is that Aristotle has a difficulty with how the form could come to be in our minds. In the Physics, he gives an account in which any sort of motion is a movement of a thing from potential to actual (or a falling away: a house can move away from being a house by collapsing, so that it is again only a potential house).
So if the form comes to be in our minds, it must have already existed there potentially. That's a very interesting claim, but it is a claim that makes sense of the idea that there is a relationship between us and the world. It's a much brighter picture than that which comes from Kant, because we really have knowledge -- the actual form of the actual things -- and it makes sense that we can convey that knowledge to others.
But then you realize that this means that all forms must exist in our minds potentially -- how could that be the case? (The claim is not as shocking as it sounds at first: if you think it through, you realize that it really must be true that, if we can have knowledge of X today, we must have had the potential to know X yesterday. Thus, it follows that you now potentially know everything that you could actually know.) It makes a kind of sense on something like an externalist picture: we are part of the world, not separate from it, and thus we are related to the world in certain ways. One of those ways could be having a mind shaped for knowledge of the world.
There is another problem, though, which is that we can also obtain knowledge through contemplation alone: for example, we can come to knowledge of mathematical truths simply by thinking. We are never encountering an actual form in an actual thing; yet we are coming to knowledge all the same. That means not only that we must have the potential for the knowledge in our minds, but that we need an account of where the actual form is that we are grasping.
Aristotle's solution is to posit an "Active Intellect," which is to say a kind of universal consciousness in which all human minds participate. This is a surprising solution, very much unlike Aristotle -- it's almost Platonic, and very similar to what the later neoplatonists will suggest. This Active Intellect contains all the forms in an actual way, and thus this explains how our minds can obtain knowledge through contemplation alone.
The modern urge is to do away with "forms" as invisible or mystical, but remember what forms are: they're organizing principles that structure matter in a particular way. These things certainly exist: this is what DNA does, for example; or, if you like, the difference between hydrogen and helium is the way in which its matter is ordered and structured. So forms are real enough; and they do exist in an actual way, and come to be in our minds when we grasp them.
Here the problem is the opposite one we had before. There are large parts of this picture that really work, and are highly satisfying; but there remain some troubles we have to sort out. Let's stop here and talk it through.
77 comments:
I don't have a fully formed response yet, but a few thoughts-
" One of those ways could be having a mind shaped for knowledge of the world."
This brings to mind the 'Tree of Knowledge, Good and Evil', and the implication that from that moment, we had within us the capacity to understand the world. It parallels this well. Actually, it predates it.
"That means not only that we must have the potential for the knowledge in our minds, but that we need an account of where the actual form is that we are grasping."
Could we not say that we understand from examination of the things around us that there are systems of order, and once we developed a language to describe or model these systems, we could extrapolate from them other systems, not found in the world (and so you posit that we "come to knowledge of mathematical truths simply by thinking"). I would argue that it is not simply by thinking that these things come to be, but that their form must lie in the observed systems we have derived these mathematical ideas from. This would be an extension of Aristotlian imagination.
"The modern urge is to do away with "forms" as invisible or mystical, but remember what forms are: they're organizing principles that structure matter in a particular way. These things certainly exist: this is what DNA does, for example; or, if you like, the difference between hydrogen and helium is the way in which its matter is ordered and structured. So forms are real enough; and they do exist in an actual way, and come to be in our minds when we grasp them."
But the form is the organizing principle, not the physical structure or 'blueprint' as in DNA- after all, what is the form of DNA? (and so on). It must by definition be a non-physical thing if we're accepting the idea at all, it seems to me. Principles are not physical.
That's it for now.
Douglas,
The Tree story isn't quite appropriate, because it was only supposed to have delivered moral knowledge. In order to understand the prohibition against eating its fruit, Adam would have had to have had a mind shaped for the world already: he would have had to have understood a number of concepts such as "tree," "fruit," and that this tree was different from other trees. (And of course, in the story, they have a fully developed language already: so, that's cognition at a very high level.)
Could we not say that we understand from examination of the things around us that there are systems of order, and once we developed a language to describe or model these systems, we could extrapolate from them other systems, not found in the world...
Aristotle would certainly be happy to say that, but there's a danger: it's only too likely that we'll extrapolate to systems that aren't found in the world. He spends a great deal of the early part of the Physics, for example, disputing the idea that motion is impossible.
The question we are asking about is how it is possible to have knowledge of the world. If we come to incorrect principles (e.g., "motion is impossible"), we haven't developed knowledge of the world. We've developed something, but not knowledge of the world.
Aristotle's model is a little alarming because he is (for contemporary minds) all too ready to find order in the world. He has some pretty strong arguments, so it's not surprising that he draws strong conclusions from them: but the conclusions are indeed very strong. For example, he argues that nature and human arts are alike in having final causes -- purposes, that is. There's a reason we can discern for everything, from why there is a squirrel to why our neighbor chose to build his house in such-and-such a way.
The contemporary conception is that we can discern a reason for why the neighbor built the house, but that the squirrel is the result of random mutations (which then happened to prove successfully adaptive -- but adaptive to a world whose life also developed in a random way). The appearance of order is supposed to be an accident, after a fashion: if we can speak of a logical principle behind it, it's the principle of natural selection.
Speaking of which:
But the form is the organizing principle, not the physical structure or 'blueprint' as in DNA- after all, what is the form of DNA? (and so on)
Very good! That's right: the problem with a hylomorphic system (that is, a system with a form/matter distinction) is that you end up with forms of forms and matters of matters.
Nevertheless, I'm not sure the hylomorphic concept is disposable. There really is nothing to distinguish the different kinds of matter (or energy, as far as that goes) besides something like form. What's the difference between red light and green light? Wavelength. What's the difference between hydrogen and helium? Atomic structure. What's the difference between hyrdogen and red light? Again, it's the way the stuff is organized: and the law of conservation of energy and matter, as well as Einstein's E=mc2, both tell us that there's a sort of basic unity between energy and matter (in the modern sense of the term "matter").
So yes, the DNA in a sense serves as my form, but it has its own form. Aristotle would have said, though, that my form is my soul: and this seems right as he means it, i.e., where the soul is an animating principle. As we were discussing below, my dead body will have the same DNA: but it will lack the animation. It is that animating principle (whatever it is) that gives me the actuality of the order that I have: it's what lets the DNA do its job, and what causes me to order my day such that I obtain meals at appropriate times (so as to continue existing, but also so that I am well-enough fed to think rationally about the problems of the world).
Thus, whether we want to say that my final cause is the animal final cause, or the Aristotelian final cause, I seem to have it from my soul (again, whatever this animating principle turns out to be). The DNA is the form of my body, but it's not my form -- at least, it's not my only form.
Well, you've hit on an interesting thing there with the mathematics, and that is what is 'real'. Are numbers real? That is, I have 2 pencils on my desk. The pencils are there. And there are two of them. Is that 2 real?
That may have connections with that 'potentiality' that you spoke of, which reminds me of a concept in certain programming languages, where the output of functions depend on the input. That is, if you think of numbers that way, 2 is a form (of sorts) and you can have 2 pencils, or 2 dogs, or 2 cars or 2 atoms. But then those things are all countable tangible objects or things. And you can have 2 thoughts, which is something less tangible, but still countable.
Or is it?
That points to a major division in Greek thought.
The Pythagoreans believed that numbers were real, and in fact the principles of all things. This faith in the ultimate rationality of the universe was such that Pythagoras is supposed to have drowned one of his followers who proved the existence of irrational numbers.
Plato is an ally of a sort to this tradition, in that he considers the numbers to be forms. Platonic forms do exist in a metaphysically real way: in fact, they are more real than the things we encounter in the world. Thus, "two" would ultimately be a form in which your pencils (or dogs, or thoughts) participate.
Aristotle didn't like this approach. Nevertheless he has a lot to say about mathematics. His approach avoids the mysticism of Plato or the Pythagoreans (which may be an advantage in the eyes of modern thinkers), but at the cost of having to deal with numbers in several different ways.
In the case of "a group of two pencils," he would like to say that the number does not exist of itself. In Aristotle's world there are things, and there are attributes of things. The pencils are things. The number is an attribute of the things -- following his system of categories, the kind of attribute it is would be a "relation." Each of the pencils has the relation of being "a member of a group of two." Likewise, you have an relational attribute of being "the owner of a group of two pencils."
It's a little strange to say that a relation of two-ness should be shared by three parties, though we have just said that: both pencils and you have the relation that makes them a pair of pencils. Likewise, if I show up and observe them, then I have a relation: "Observer of a group of two pencils." And so forth.
The other problem is that sometimes -- very often, in fact -- number proves to be a crucial feature of the form of a thing. Aristotle offers the example of an octave, in which the mathematical relationship defines the thing. In this case there is no getting away from the fact that number has a substantial quality -- and that is going to hold for anything else defined by number (as in the hydrogen/helium example).
Thus, it seems better to me to say that numbers are real in something more like the Platonic way -- even at the possible price of mysticism. Not that Aristotle wholly avoids it, since (as mentioned) he ends up with an Active Intellect that holds actual versions of all of the potential concepts.
By the way, I think I mentioned this a few months ago, but a fellow I know is a philosopher who specializes in mathematics. He tells me that the main debate in philosophy of math for the last century or so is on the status of the infinite. That's a subject that Aristotle argues about at length in the Physics; but apparently it's just as live a topic today.
So the question of whether numbers are real -- especially infinite series -- remains alive.
We should probably explore this a bit more:
But the form is the organizing principle, not the physical structure or 'blueprint' as in DNA- after all, what is the form of DNA? (and so on). It must by definition be a non-physical thing if we're accepting the idea at all, it seems to me. Principles are not physical.
In a way that isn't quite right -- causes are principles for Aristotle, and some causes are physical. Why did the billiard ball go that way just then? We have four causes in Aristotle: final, formal, material, and efficient. Two of these -- the material and the efficient -- will be physical, and the formal cause, which has to do with the structure of the billiard ball, is both formal and physical: the matter is structured by the form, but it is still matter.
Thus only one of the causes will prove not to be physical: the final cause, which is the playing of the game of billiards. But this cause has another cause, which is the player: and this player has causes that have caused him to come to be, some of which are physical and some not.
So we can't say that no cause is physical (and, since causes are principles, we can't say that no principles are physical). But there are certain causes of motion that have to be non-physical for Aristotle.
Aristotle argues for this in Physics Eight, chapter five. Let's say you have a motion: a man hits a ball with a bat. The cause of the motion of the ball is the bat; the cause of the bat the man. The man moves himself.
But can we say that he moves himself? His hand is moving the bat, but it is moved by the muscles and structure of the arm, which are linked to the muscles of the waist, which... you get the picture. Insofar as it makes sense to talk about this movement as a whole, it is caused by the man (by his soul, which is playing the game); but if the movement of the whole is what counts, the movement of the parts is accidental (i.e., the way a nail in a ship moves not of itself, but because the ship moves). That's not right.
Thus we have to come to some part of the man that can move as a whole. What kind of thing is that? Well, it has to be a thing that doesn't have parts: if it had parts, we'd have to ask which part was moving the other part, and so we'd have an infinite regress.
If it has no parts, then it isn't material, for all material things have parts (at least a beginning and an end). If they don't have parts, they lack extension; and if they lack extension, they can't move (because to move would mean that there was a beginning part followed by an ending part of the motion).
Thus, the kind of thing that can move itself is immaterial: it isn't physical in the sense you mean it. That's the kind of principle that Aristotle thinks has to underlie something that moves itself, as opposed to being moved by something else.
But notice what we've said about it: it has to move itself as a whole, but it cannot move. So what Aristotle ends up saying about it is that it is an "unmoved mover" -- it causes motion without itself moving (since movement for an immaterial thing is impossible, as it doesn't have a beginning or an end).
That's really mysterious. How does it move a physical thing without being physical -- and without moving or being capable of motion?
Let's talk some more about that, and also about Eric's proposition regarding mathematical objects. Here's a question: which is prior, the first or the second dimension? The second or the third?
Let's say you have a motion: a man hits a ball with a bat. The cause of the motion of the ball is the bat; the cause of the bat the man. The man moves himself...Insofar as it makes sense to talk about this movement as a whole, it is caused by the man (by his soul, which is playing the game)...; but if the movement of the whole is what counts, the movement of the parts is accidental (i.e., the way a nail in a ship moves not of itself, but because the ship moves). That's not right.
Thus we have to come to some part of the man that can move as a whole. What kind of thing is that? Well, it has to be a thing that doesn't have parts: if it had parts, we'd have to ask which part was moving the other part, and so we'd have an infinite regress.
No, and no, and no again.
Firstly, there is not an "infinite" number of "parts" in a human being, no more than there is an "infinite" number of particles in the known universe. (In fact, considerably less. 7 x 10^27 atoms in the human body, says Wikipedia.) No "infinite parts" means no "infinite regress." (And given modern cosmology, this is true even if you take it back to the Big Bang.) It might've made sense in Aristotle's day to think this - it makes none in our own.
Whether it is the movement of the whole that "counts" depends on why we're considering the motion at all. (If the issue is whether this will exacerbate an arm injury, or break the bat, we're only interested in certain parts of the motion; if the issue is whether the nails on the ship will pop out under ordinary stresses, then the ship's destination is less important than the movement of the individual plank). Thus, it can't be right to say "only the motion of the whole counts, period," in some universal sense of "counting" - or that the movement of any part is "accidental." How does "accidental" contrast with "counting" anyway?
The answer, I think, is that you're saying something "counts" if it is done with a conscious purpose. (Though these threads seem to have a lot of blurring between the kind of conscious purpose a thinking being can have, and the metaphorical "purpose" we assign, as a convenience of speaking, to various other things.) But this says nothing about the mechanism by which that conscious purpose is formed - and it gets us nowhere near why a "thing without parts" has to be forming that purpose. The remainder of the argument, pointing to an immaterial soul, then doesn't follow.
Here's a question: which is prior, the first or the second dimension? The second or the third?
Here's a question - what does that question mean? (I'm not being flip, really I'm not.) If you mean dimensions as they're used in mathematics - whenever I've dealt with a problem that uses more than one, they're all present in the problem, and "priority" isn't raised. If you mean spacial dimensions - then as far as I know they all exist all the time; so what is this "priority" you're asking about?
Eric, as I understand him, was answering -
There is another problem, though, which is that we can also obtain knowledge through contemplation alone: for example, we can come to knowledge of mathematical truths simply by thinking. We are never encountering an actual form in an actual thing; yet we are coming to knowledge all the same.
And I think his question is on the right side of it. The basic positive integers, and the properties we assign to them, are based on the behavior of things in the physical world. (Which is why putting one pencil together with another gives you two.) At that level we are very much in contact with the physical world; and without that kind of evidence, we wouldn't have the foundations of modern mathematics.
But go further - manipulate these numbers by the rules of algebra. Make some assumptions about infinity and prove the fundamental theorem of integral calculus; develop mathematical analysis further, to create probability theory and statistics. When you apply the results to the physical world, they match. The Central Limit Theorem - I think we talked about it briefly once before - posits that Gaussian distributions (i.e., bell-shaped curves) are often going to arise when you're dealing with averages -- and this just what happens, whether you're throwing dice or measuring IQ scores. So these mathematical truths, at least, even if they are derived through pure reason, begin and end with the outside physical world and how it behaves.
...and this is important, because it means we really are not obtaining knowledge through contemplation alone. (Which appears central to the main argument.) The "contemplation" starts with something we got from outside.
No "infinite parts" means no "infinite regress." (And given modern cosmology, this is true even if you take it back to the Big Bang.) It might've made sense in Aristotle's day to think this - it makes none in our own.
There are two things to say about that.
1) The infinite regress isn't the problem; it's just a difficulty that comes up along the way. The problem is the question of how motion gets started. Let's say there isn't an infinite regress, but that we come to the first mover in the sequence. Does it move itself, or is it moved by something else?
It's not moved by something else, since we've stipulated that it's the first. Since it's physical, it must have extension (and it is thus theoretically divisible into parts, even if it is not actually so divisible -- this is the old Greek atomist position, by the way, against which you will find the Greeks have a number of logical arguments).
We want to know how this thing moves itself. It has to move as a whole, so that means that its theoretical divisions ('front' and 'back' or whatever) move accidentally: it essentially moves as a whole. But what causes this motion? Itself, yes; but what is the causal mechanism?
I think Aristotle is wrong, but the argument is very troubling. We end up needing to say that there isn't a first mover (in the case of me moving my arm) because it is interactions that create the movement, not a sequence. Yet the interaction has parts, and also starts somewhere: we need to come down to a part that moves itself rather than being moved by something else.
2) Before we settle in on a question of whether reality is infinitely divisible or not, we have to distinguish between types of divisibility. Aristotle doesn't believe in actual infinites either: he's quite clear on the point in the Physics.
What he believes in are potential infinites. In terms of actual smallness, the atomists believe as we do today -- that there will come to be some smallest part beyond which no more division is possible. (Likewise, Proclus argued the same for time: there must be a 'time atom' that is the smallest possible division of time.)
On the other hand, we need infinite divisibility to make sense of mathematical objects -- with any continuous length, we can talk about half of it, and then half of that, so on forever. This kind of infinite (the potential infinite) is something that Aristotle believes is both real and necessary.
So when he is looking for a first mover and comes to the conclusion that it must be unextended (and therefore not physical), he isn't asserting that the smallest particle can be effectively further divided. He is talking about the logic of dividing it mentally: the front and the back, the beginning and the end, and so forth.
On mathematics:
If you mean dimensions as they're used in mathematics - whenever I've dealt with a problem that uses more than one, they're all present in the problem, and "priority" isn't raised.
It's raised here because it also pertains to the question of motion. Zeno's paradoxes about motion seem to arise because of a priority mistake about spatial motion. Aristotle's rebuttal is that Zeno is assuming that a line is composed of points, rather than being divisible into points. (Likewise, that time is composed of a collection of nows without extension, rather than being divisible into them).
Thus you get the paradoxes: if we can divide distance into a point that has no extension, we have to cross an infinite number of such points to reach some later point. It isn't possible to go through an infinite series without infinite time; thus, no motion. (The other paradoxes are variations of this; the third, regarding the frozen arrow, is my favorite).
Aristotle's response is to say that the line has priority over the points, and time over the moment; and indeed, that the solid has priority over the plane (because that is what really exists in the world -- here we are not worried about Kant's problems, but ready to accept reality as we find it).
We get the concept of the plane by imagining we're only dealing with the surface of the table (i.e., making a mental and theoretical division of the table in the same way we were talking about theoretical divisions above). Then we get the line by making a further division of that table into 'just the line from here to there'; then we get a point by dividing the line. But the priority goes to the three-dimensional object, Aristotle says, because it has reality in a way that the others do not: it is an actual object, not a theoretical division.
Since you also wanted to talk about the smallest actual object, any further division of which should be theoretical (especially into points, and especially because you wanted to avoid a paradox about motion!) you should be quite sympathetic to Aristotle here.
So these mathematical truths, at least, even if they are derived through pure reason, begin and end with the outside physical world and how it behaves.
Good! I knew you would like the ancients better; this is just the kind of argument that Aristotle would make.
So we want to say that our mathematical concepts must be real even where they are not rational -- because they line up with reality. Notice we don't escape Kant's problem here: he would say that they line up with what our mind is prepared to represent to us about reality -- and of course they do! The mind is using the same toolset to construct both, he would argue.
But here is one place I think we may get a thin wedge against Kant: some of these mathematical facts we come to (like Pythagoras' unfortunate follower) are irrational, but they still seem to be true. What do you say about this?
Some things are defined in mathematical semantics as irrational, but whether or not something is rational would depend on the system it's being defined within, would it not? Rational is sometimes defined as being not only reasoned but optimal- optimal to what? That is the question. Since almost all numbers must be irrational numbers, perhaps the universe's underlying rationality is quite different than ours neat and clean mathematical system or reason.
"The Tree story isn't quite appropriate, because it was only supposed to have delivered moral knowledge."
I would say value knowledge. Prior to partaking, Adam would not have had a full knowledge, and therefore could be said to not have the potential for the knowledge of all forms in our minds- rather like animals, I think. That tree gave the scope and type of knowledge that separates us from animals- a value scale, which lead to morality and therefore also philosophy.
"it's only too likely that we'll extrapolate to systems that aren't found in the world. He spends a great deal of the early part of the Physics, for example, disputing the idea that motion is impossible."
Well, of course- we are only human after all- once imagination grabs something, we may or may not imagine things which could be correct. We make mistakes, as must be true of any imagination- and we must make errors, as these are at times the best instructors as to where the truth/best solution really lay. I fail to see why this is a danger or a problem.
"The appearance of order is supposed to be an accident, after a fashion: if we can speak of a logical principle behind it, it's the principle of natural selection."
Ah, but that may be our imaginations at work, and one of those mistakes I just mentioned- we cannot know that yet. It may also be correct. Curiously, I tell my kids that accidents have causes- they aren't 'just accidents'. I may be wrong on a grand scale, but perhaps not. Let's consider the case of natural selection- every mutation that forwards evolution is an 'accident' but it's a mutation (which have identifiable causes) and one which survives, therefore one with purpose (furthering survival of that specie). Curiously, evolution as described by the theory, is progressive- single cells develop into multi-cellular creatures, which develop into more complex creatures and into insects, then vertabrates and so forth. Progression implies an order- can randomness posses underlying order? If yes, what then would 'order' mean?
"Aristotle's response is to say that the line has priority over the points, and time over the moment; and indeed, that the solid has priority over the plane (because that is what really exists in the world -- here we are not worried about Kant's problems, but ready to accept reality as we find it)."
I would ask him then, if what is present in the physical world has priority, does our body have priority over our anima/intellect/soul? Shouldn't it be the other way 'round, as you cannot have a solid without a plane, a plane without a line, or a line without a point? Isn't the point the beginning of the description of the physical universe?
Douglas,
...whether or not something is rational would depend on the system it's being defined within, would it not?
The point here is that 'the system it's being defined within' is our system. If Kant were right, and all we have of the outside world is only our self-generated phenomena, we shouldn't be running up against things that don't compute with our mental architecture. Our system should be processing these details in a rational way or, if it cannot, failing to handle them.
Instead we seem to be able to recognize them, and to learn something important about the world outside of our minds. For example, one thing we learn (from Georg Cantor) is that almost all real numbers prove to be irrational.
But another thing we learn is that, while "almost all" numbers are irrational, the set of irrational numbers is exactly equal to the set of rational numbers. The proof is simple: start with any irrational number (say the square root of two). Pair it with a rational number, say 1. Now pair the next irrational number with the next rational number. Since both sets are infinite, you will never come to a point at which you cannot continue to pair the two sets: this means they must in some sense be exactly the same size.
We can't properly conceive of how to hold those two propositions at the same time (i.e., that the first set should be both vastly larger than the second, and exactly the same size). An idea that isn't properly conceivable by a human mind can't be a product of that mind -- unless, of course, it represents an error, a malfunction of the machinery.
But what is interesting is that we can take some of these very irrational features, and use them to produce startlingly accurate physical calculations.
Real objects cannot have infinite charge or mass or whatever. But when scientists in the 1950s started calculating those quantities with their latest and fanciest theories, infinities kept sprouting up and ruining things. Rather than abandon the theories, though, a few persistent scientists realized that they could do away with the infinities through mathematical prestidigitation. (Basically, they started calculating with and canceling out infinity like a regular old number, normally a big no-no.) No one liked this fudging, but because it led to such stunningly accurate answers, scientists couldn’t dismiss it. In fact, the reigning paradigm in physics today—which describes the workings of invisible “fields” (similar to magnetic fields)— would not exist without this hand waving. And now physics is stuck with fields: they’ve become more fundamental to understanding the universe than mass or charge. Fields have become the very fabric of reality—even if our understanding of them relies on some unrealistic assumptions.
This is a reason to believe that mathematics is real, i.e. a feature of the universe rather than our minds. Our minds clearly aren't creating this aspect: they can't have created it because they can't even properly conceive it.
This seems to me to be the proof we are looking for that the skepticism is wrong: our experience of reality isn't an invention of the mind, but does in fact represent a relationship between ourselves and the things we know about.
But Aristotle is also wrong: we don't know a thing by knowing its principles (so that we can give a rational demonstration from the principle to the thing, i.e., from cause to effect). The best demonstration that we know a thing in the world is this demonstration from the irrational and the not-properly-conceivable. This is what most clearly shows that we have a relationship with the real thing in the world, which is the kind of thing we have come to believe knowledge to be.
I don't think that I deserve any credit for proposing knowledge as a form of relationship, really. That's the direction I was moving toward, I think, but I still don't fully understand what that means.
As for mathematics, it may begin in the real world, but it does not always end there. Theoretical mathematicians don't seem to even try to relate what they do to the world we live in. Sometimes their work ends up being useful, but that's not what they're trying to do. And, sometimes their work doesn't.
As for mathematics (and science, for that matter) modeling the world, that too is problematic.
Let's look at Aristotelian-Ptolemaic cosmology. The earth was an unmoving body at the center of the cosmos and the heavens spun about it. Ptolemy developed quite a sophisticated mathematical model that allowed accurate predictions of where various heavenly bodies would be, eclipses, equinoxes & solstices, etc. It also explained why solid objects put into the air fall toward earth, why air bubbles rise in water, why fire rises in air, meteors, etc.
It all worked: it fit what our senses perceived, it was all modeled mathematically and was predictive, and it made sense of how the entire cosmos worked. If you are a pragmatist, then it was pragmatically as true then as physics is today. And, it lasted more than a millennium before it was seriously challenged.
When Copernicus proposed his heliocentric system, he had made no new observations to suggest one. (Such observations couldn't have been made in any case until the invention of the telescope allowed astronomers to see the phases of Venus.) In fact, he wasn't much of an astronomer at all; he was a mathematician. His system was no more accurate than the Ptolemaic mathematics. He proposed it because it was more mathematically elegant than the Ptolemaic mathematics (it did away with the equant, an inelegant necessity of the Ptolemaic system).
And then, while Galileo's observations challenged the Ptolemaic system, they did not necessarily suggest a heliocentric cosmos. Tycho Brahe, who had far and away the best observatory on the planet at the time, reconciled Galileo's observations with an alternate geocentric system, which also worked out quite well mathematically. In addition, it didn't upset Aristotelian physics.
And yet, all of those centuries of carefully recorded observations, painstaking mathematical modeling, predictive usefulness, confirmation in physics, etc., turned out to be quite wrong in explaining what was really happening.
NB: I hadn't yet read Grim's 2:12 post when I was writing mine. Mine is not a response to his.
Yes, but it's a great parallel comment. Obviously, we seem to be getting somewhere. Which of course only begs more questions. Well, that's philosophy, isn't it?
As to the issue of canceling out infinity, I think first we'd have to address what infinity is- in short hand common usage, we use it as a term of -going on forever, endless. However, what is the difference between that and -extending beyond our capacity to resolve? From our vantage point they are functionally the same. Philosophically, they are quite different. Something could go beyond our capacity to resolve, and yet be finite in some greater sense. Or perhaps there are two different things that we both categorize as infinite- some which are and some which are not. If infinite is only one of these it either creates the paradox of rational numbers and irrational numbers being equal yet not equal, or it creates a finite (but beyond our capacity to resolve) set of real numbers or both sets are finite. If there are two flavors to infinite, then perhaps something can also be both...
The problem is the question of how motion gets started. Let's say there isn't an infinite regress, but that we come to the first mover in the sequence. Does it move itself, or is it moved by something else?
Just the sort of question that...no longer makes much sense in the light of modern physics. If the Big Bang is correct, by the end of the Planck Epoch you have things already in motion, which motion hasn't stopped since, and which sometimes takes forms Aristotle couldn't have imagined.
Aristotle did not have even Galilean relativity at his disposal - I've read, anyway, that he believed instead in "impetus" ("continuation of motion depends on continued action of a force" - contrary to Newton's First Law, which describes the world better) - you'll correct me if I'm wrong on that.
As I understand it, the relevance of this topic to what we're talking about is that Aristotle came up with the idea that the human soul was "originating" motion - rather than being composed of parts that were already in motion. It's understandable he'd have such an intuition, and an elaborate argument for it, but there's no reason to credit this idea today - or at least none I have seen. (Actually, this fellow has an argument that may run that way - I do not understand it well enough to have an opinion about it - but the arguments over that use tools that were not available to Aristotle, and are not informed by the arguments you're talking about here.)
Just the sort of question that...no longer makes much sense in the light of modern physics.
Actually, it makes just as much sense in terms of modern physics -- unless you adopt a hard deterministic perspective. Unless all motion tracks necessarily to the Big Bang, that is, we still need to sort out how it is that motion gets started.
If I'm standing in a field, and I'm looking up, and I see a ball is coming, what's the process by which I begin to move to catch the ball? It won't do to say that 'that question doesn't make sense, since motion has always been around since the Big Bang,' unless we want to take a pretty radical view of the playing of baseball.
The logical problem Aristotle is describing exists any time we want to account for a kind of thing that moves itself, instead of being moved by something else. It's not a question of Aristotle's ignorance of modern physics -- it's a question of logic.
On Zeno -
Thus you get the paradoxes: ...we have to cross an infinite number of such points to reach some later point. It isn't possible to go through an infinite series without infinite time; thus, no motion. (The other paradoxes are variations of this; the third, regarding the frozen arrow, is my favorite).
I liked the falling egg that never hit the floor, myself. (As it passes through ever smaller units of space, it takes ever smaller amounts of time to do that...) But anyway -
That was a weakness in Greek thinking, great though it is - they couldn't handle the idea that an infinite series can sum to a finite sum. Yet it can, and in modern mathematics, it does. That was a change of assumptions, but it has this advantage: it models the real world we observe, and not the fantasy world of Zeno's paradoxes. (Who was the king who dropped a bowl to the floor, and said he'd refuted the paradox? Good empiricist, that man.) If you don't accept that you can sum an infinite series, you can't get integral calculus...nor that which comes after it. Yet if you do change the assumptions, and use integral calculus...you get mathematical models that describe the world we observe very well indeed.
Not just the egg that falls and makes it all the way to the floor, or the arrow that sinks into my flesh without stopping...but the motions of the planets, and the spectra emitted by excited atoms. That's what I was talking about before - the way mathematics isn't just a matter of arriving at truths by pure cogitation, but at the beginning and also later, is informed by contact with the outside world. So if Zeno's paradox is answered by this change in starting assumptions, a change that is required by the world we observe, where does that leave the problem you're dealing with?
Euclidian geometry - and I do know something about that - derives from a fixed set of postulates, most of them explicitly stated in book I of the Elements. It gives the enterprise a wonderful rigor - but it can lead to dangerous habits in thinking, if you're modeling the real world, and start to think your postulates may never change. (I'm guessing you're familiar with the way non-Euclidian geometry was created...and saved from obscurity by the fact that it proved useful for General Relativity.) Good Euclidians had no use for infinite steps - that is why they posed the "circle squaring" problem the way they did (and it ultimately proved impossible under their rules and assumptions - though it's trivial once you allow infinite steps or approximate solutions). But you have to throw away their aversion to infinity to get closer to the real world we can measure and make calculations about.
Actually, it makes just as much sense in terms of modern physics -- unless you adopt a hard deterministic perspective. Unless all motion tracks necessarily to the Big Bang, that is, we still need to sort out how it is that motion gets started.
#1, well, what's wrong with a hard deterministic perspective? It could even be correct!
#2, And if it isn't true, you won't find out by the kind of reasoning you're using here...the "sorting out" will need the tools of physics, and not simply introspection about how your own mind works.
If I'm standing in a field, and I'm looking up, and I see a ball is coming, what's the process by which I begin to move to catch the ball? It won't do to say that 'that question doesn't make sense, since motion has always been around since the Big Bang,' unless we want to take a pretty radical view of the playing of baseball.
That's because you're taking a process we normally think about as a psychological one - the mental processes that cause you to react to the baseball - and trying to recast it down to the level of physics. That is a pretty "radical" thing to do, but if you do it, the answer I'm giving is not especially strange.
It made a lot more sense for Aristotle - he knew nothing of cell structure or biochemistry, through no fault of his own; and had no reason to imagine just how complicated these things are.
To use the familiar "computer" analogy, if a programmer asks you how your program performs Gaussian elimination on a matrix - or, let's keep it simpler, carries out the Euclidian division algorithm - you'd give him an answer in terms of its subroutines. This subroutine divides everything in the first row by the first number; and this subroutine multiplies the result by the first number in each other row, and subtracts the first row from each row; etc. etc. If the person demanded to know more, you might show him the specific commands that made up each subroutine. If he demanded to know more, you might show him how your high-level programming language is broken down into machine language, the commands actually recognized by each chip. If he still wants more...you might eventually have to reach a point where you are teaching him about the flow of electricity and semiconductor physics. The problem is not that you are being radical. The problem is that he is posing the problem in a "radical" way.
The Greeks weren't quite as weak as that: they understood that aspect of infinity quite well. Plato refers to it as 'the great and the small,' as does Aristotle in referencing Plato. The point was that you get the infinity by division, and thus there is an infinitely large number of possible divisions ('the great') that produce infinitesimal proportions ('the small').
So yes, you can get a finite sum from addition of an infinite series. The problem Zeno is bothered by is that you can't get extension out of any series -- even an infinite series -- of addition of things without extension.
I'm not sure if that is a fault of Euclid's (or his predecessors) in terms of how they specified their terms, or not. It may be that the concept of an unextended thing simply doesn't make sense as a real physical fact; if all of creation is extended, then the point is merely an intellection (and note that this is exactly equivalent to Aristotle's emphasis of the priority of the third-dimensional, actual spatial object over the point we are imagining).
But we've reached what seems to me to be a useful floor, and we've found it in an interesting place. It seems like we are agreed that the world doesn't fit our internally generated forms; but when we allow the math to 'be,' so to speak, and follow it to conclusions we cannot always even properly conceive, it produces accurate results.
That seems like the solid ground we have been looking for: a way of contesting the idea that we are trapped in representations and phenomena. It seems that we are getting some data about the noumenal world in with our representations -- and we found the proof of it not where our world view is coherent with our observations, but just where our world view seems most upset by our observations. It's the places where our thoughts force us to confront things not properly conceivable that we find our best evidence that we have real knowledge of the world outside our mental systems.
That strikes me as a good place from which to regroup, and move on.
And this, in a way, goes back to what I was saying about mathematics. Physics...even seventeenth-century physics, let alone modern physics...got so much further than Aristotle because it kept getting back in touch with the evidence, and not simply reasoning out from the facts that were available to him. I've read that some Greek astronomers thought that heavenly objects obeyed different rules than objects on earth...and that was a logical view, from what they knew...it took many observations they had, and advances in mathematics that were the work of genius, to show that (for example) the planetary orbits could be explained with a simple law of gravitation that works right here on the earth. This isn't to disparage the logic of the Greek astronomers...only to say that, once better evidence came in and better tools were available, logic pointed a different way.
It's not that heavenly objects obeyed different rules, but that the objects were thought to have different natures. Rocks fall down when you let them go; for Aristotle, they sought the center of the universe until they were stopped by the intervention of some other object (like other rock). Stars were supposed to be made up of stuff that moves not down, but in a rotational fashion. It wasn't obeying different rules, but rather, both different kinds of things were obeying their different natures. The distinction is significant, because it's a major breaking point between early and modern scientific thought.
Before the late middle ages, physics assumed that to know how a thing would move, you needed to know what kind of thing it was. That is, you needed to know its nature. A pound of rocks moves differently from an equal weight of water, all things considered.
The idea that there were universal rules governing motion begins to appear in late medieval thought -- Gersonides is an early thinker who begins to posit such rules. It's problematic in a way, because rocks and water really do move differently -- you end up needing a lot of explanation of just why water doesn't move the same way as stone, if there is a universal law governing both.
This concept of universal laws of motion is so fundamental to Newtonian and post-Newtonian physics that it takes quite a bit of careful thought to understand what it was like not to have it. If you want to understand Aristotelian physics -- and you ought to want to do so, because it's full of dangerous questions that ought to provoke you to explore what you think you know about modern physics more carefully -- this is one of the hard leaps to make.
So we want to say that our mathematical concepts must be real even where they are not rational -- because they line up with reality. Notice we don't escape Kant's problem here: he would say that they line up with what our mind is prepared to represent to us about reality...
But Kant's problem still strikes me as a non-problem...simply demanding a level of certainty we can't have, and declaring if we don't have that, we have nothing at all. I'm sick in bed, a friend comes to visit. We start talking about geography. I show him a map of the Iraq-Saudi border and talk about what a threat Saddam's regime was to the Gulf states, with this wide, flat border he could roll across anytime he liked...
He responds, "Shut up. All you have is that map. You haven't experienced the border directly, and you're bed-ridden, so you can't get there. So you know nothing.." I point out to him that various other maps agree, and so do satellite pictures. He says, I can't prove that all those maps and satellite pictures are not the result of a vast conspiracy to deceive me about the Iraq-Saudi border, which is in reality fifty feet long and in the middle of a wet, soggy swamp. I talk about reports from troops who have guarded and crossed that border - and he tells me, "For all you know, they're in on it! The whole world has been arranged to deceive you, and you can't prove it isn't, so you have no knowledge. So just shut up."
Has Kant given me the slightest reason to think that my mind is so deceitfully arranged, that nothing in it corresponds to the world outside? It could just possibly be true, but admitting that possibility - you must stop there, because there's nothing to do with that possibility. So you either paralyze yourself for fear you don't have certainty - or you throw it aside and don't worry about it. You assert to me that it's "wrong" for me to approach this problem this way. But you don't show me why. If Kant's "problem" is nothing more than this, then why do I need a "thin wedge" against it from this discussion we've been having?
(Now as it happens our minds really are good at rationalizing and systematizing things in ways that do not fit reality. That's how we got Marx and Freud, and magical thinking, and self-deception, and a great deal more besides...and one reason we must be careful about our intuitions, and try to stay in contact with the evidence, nor have too much confidence when our reasoning carries us far away from it. But Kant, as you describe him, is carrying this to a level where there's no evidence.)
[S]ome of these mathematical facts we come to (like Pythagoras' unfortunate follower) are irrational, but they still seem to be true. What do you say about this?
I don't see what's "irrational" about them. We have to change basic assumptions - the ones Euclid or Aristotle worked with - in order to accomodate them; but once we do, we reason about them just the same as we reason about the facts we accepted before.
(The term "irrational fact" makes no sense to me. "Irrational thought process" or "irrational conclusion" I can understand; "irrational fact" I do not.)
So you either paralyze yourself for fear you don't have certainty - or you throw it aside and don't worry about it.
This is what I meant when I said that a common response among practical men was simply to pretend the problem doesn't exist. It's a problem that doesn't make any difference in terms of what you're going to do; so why bother with it?
Notice, though, that we've gotten some good out of bothering with it. A moment ago you might have been satisfied asserting against this impractical objection that it made no difference, so why bother? But now, you have some real grounds for asserting that actually, in point of fact, Kant's model is the one that makes no sense. Even given the problem of apperception, we do get noumenal input in our phenomena.
As far as I know, no one has ever defended the model we are formulating here. These problems we started with are standing problems for philosophy, epistemology, and science.
What I think we've done in the last few days is build something new. If I'm right about that -- I will pursue the matter and let you know -- then this represents not just an easing of your doubts, but an advance in an argument that has involved many of the greatest minds of human history
So you still end up taking up the sword and fighting Dr. Foul. It's just that now, you are relieved of the worry that you might (in some metaphysical sense) be fooled. You can wield the sword with confidence. Furthermore, in addition to taking sword against Dr. Foul we've slain a demon on the way.
I don't see what's "irrational" about them. We have to change basic assumptions - the ones Euclid or Aristotle worked with - in order to accomodate them; but once we do, we reason about them just the same as we reason about the facts we accepted before.
(The term "irrational fact" makes no sense to me. "Irrational thought process" or "irrational conclusion" I can understand; "irrational fact" I do not.)
What I mean is this: a fact is opposed to an opinion, in that it can in theory be tested against reality, in the way you are talking about, and proven true or false.
The fact that almost all real numbers are irrational is a fact; the fact that, nevertheless, the two sets are of the same size is a fact (and a provable one, the proof being given above).
This is a plain violation of the law of non-contradiction: the two sets are both the same size and not the same size. Anything that violates the law of non-contradiction is irrational. Nevertheless, this is a fact; and it is therefore an irrational fact.
Thank goodness it is! It is with these irrational facts that we find our firm ground. We might have doubted the rational facts of the world as products of a Kantian representation; but we cannot believe that the irrational facts are such things.
To say that the law of non-contradiction does not apply in certain places, by the way, is more than a change of assumptions! It's a serious matter indeed. You'll find, when I have a moment to collect my thoughts, that it is prefigured in Plotinus.
The fact that almost all real numbers are irrational is a fact; the fact that, nevertheless, the two sets are of the same size is a fact (and a provable one, the proof being given above).
I assume you mean "irrational" in the mathematical sense, not "irrational" in the other sense here.
The fact that almost all real numbers are irrational is a fact; the fact that, nevertheless, the two sets are of the same size is a fact (and a provable one, the proof being given above).
You could've used a simpler example - the set of positive integers is the same size as the set of all integers; or the set of positive integers is the same size as the set of even integers. (These are a "countable infinity," versus the "uncountable infinity" of the reals and the irrats.)
This is a plain violation of the law of non-contradiction: the two sets are both the same size and not the same size.
Oh, no it isn't!
The concept of "size" the way Aristotle used it was pretty well useless in dealing with infinities - Greek mathematics never really got a handle on it. (Then again, they had a much weaker notation than we now have, and I'm more awed by how much they did than that they didn't get beyond it.)
For Georg Cantor to talk about different kinds of infinities, countable and uncountable (aleph-null, and aleph-one), he had to redefine "size" in a way that worked. So "size" for him was not a matter of - "count all the objects here, count all the objects there, and see if it's the same." For obvious reasons, you can't "count all the objects" in an infinity, because you never get to the end of it.
So instead, he took a different process. "Pair these things off" - that is to say, see if you can make a rule that pairs each thing in your series to a positive integer. If you can do that, you've got a countable infinity; if you can't, an uncountable. (Horribly simplified, but so it is. Gamow's One, Two, Three...Infinity, incidentally, has a picture of a Kalahari Bushman trying to handle a number greater than "two" next to Cantor at a blackboard, both examples of men trying to get a grip on bigger numbers than they knew. Charming fellow, that Gamow.)
The fact that Cantor and the moderns have expanded the concept of "size" to deal with concepts Aristotle couldn't does not mean their ideas are self-contradictory, or in any way irrational.
(This is by no means the only areas where that's been done -- the French mathematician Lebesgue expanded the concept of the "integral" to something far more inclusive than Newton or Leibniz would've understood, which allowed calculus to deal with functions they couldn't've. Not a contradiction, just an expansion of the concept. I never learned to work with it, but at least I know it's there.)
("If you can't" should be read as, "If you can pair them off with the irrational numbers"...I think he never proved, or accepted as proven, that these were the only two infinities available.)
Yes, but notice that we've just specified a rule for counting the irrationals -- by pairing them each with a rational number. Thus, the set of irrational numbers is a countable infinity.
However, the set of real numbers is an uncountable set. Cantor's own proof establishes that (as the uncountability of many subsets of real numbers, including the 'Cantor set'). The set of rational numbers is countable; since the whole of the set of real numbers is the rational plus the irrationals, the irrationals must be an uncountable set.
Thus, we're back to a contradiction. The set must be countable (because there is a rule for pairing them off, which we have just given); but it must also be uncountable (by Cantor's own proof).
(He's Lord Foul and you fight him with a ring of white gold - but that is another story. Bedtime for me. Sweet dreams!)
Oh, if you are saying that, then I have to correct you here -
[S]tart with any irrational number (say the square root of two). Pair it with a rational number, say 1. Now pair the next irrational number with the next rational number.
And what makes you think that you can determine "The next irrational number"? That's not good mathematics. (Since I really am going to bed it'll have to wait for later, if this point is important to you. But the way you can pair rationals with positive integers isn't something you can do with irrationals.)
Grim, do we not run the risk here of finding that for the universe of our own creation (Kantian) to work, it has to have errors and seemingly irrational facts in it? Didn't we just look at Georg Cantor and the theory of non-countable and countable infinities, and so isn't it therefore that we've now resolved that everything works, and so we're back in the trap of Kant being able to say 'see, you've resolved it- or course- since it's internal to you you must be able to resolve it.'.
Or am I really confused?
I wondered if you'd raise that objection; being a non-mathematician I wasn't sure if number lines are supposed to be composed of numbers in the way that lines are not supposed to be composed of points. If there is no "next" irrational number, then we're in precisely the same case as with the 'great and the small.' You cannot divide a line at a point and the point 'next to it,' because between any two you specify you can find a third (and indeed, an infinite number).
Aristotle deals with this set of problems throughout the Physics. These are not then two different "sizes" of infinities; they are exactly the two different kinds he defines in the Physics.
The issue of priority we already discussed is really sticky here. It's fine to say (as Aristotle does) that the line has priority over the points. That is merely to say that the point isn't real except as a way of designating a division of the line.
When you select a number on a continuum of numbers, though, it is highly problematic to make a parallel claim. Irrational numbers only exist in a continuum, of course, so I suppose you can clain that the continuum has priority; that makes a degree of sense. (Irrational numbers don't exist in cases like "two pencils," but they do in cases like "two gallons of gas," assuming that gas were infinitely divisible. Thus, the existence of a continuum is prior to the possibility for the existence of irrationals).
As a mental object, though, we end up with a kind of nothing that has priority over the numbers: the continuum here isn't 'gallons of gas,' but just a kind of pure potential for extension. The Aristotelian perspective was very interested in the relationship of potentiality, actuality and extension. Avicenna calls 'prime matter' a sort of pure potential to take on extension; and there's an interesting element in Aquinas in which existence is a kind of further actualization of a form by virtue of its actually taking on extension.
It strikes me that none of this undermines the claim that it's out of order with our systems to think this way. It is also not properly conceivable to say that there are an infinite number of points between two points on a line. I can understand the words and use the concept, but I can't actually conceive of it (no more than I can actually count to infinity). I think the larger point we were after holds.
Douglas:
I don't think we're in danger of that trap as long as we are dealing with things that are what I am calling 'not properly conceivable.' These are cases where we find things in the world that our brains/minds cannot conceive in the proper sense -- I can properly conceive of one or two or a hundred, but I cannot in the same way conceive of a continuum composed of infinite numbers so small they never touch, and indeed so small that between any two of them are an infinity more.
What I can do is devise a model that lets me work around the concept even though it won't fit my machinery. That's remote from the danger of my mind having constructed reality, though, because the mind is being forced to grapple with something it couldn't have constructed (because it couldn't, and indeed still cannot, properly conceive it).
My reason is not what is at work in the world I'm encountering, in this case. Thus, it must be something else. This something else can be predicated approximately by these models, but only insofar as I accept models that do things that still don't entirely make sense to me (as with the example of canceling infinities in physics). That suggests we are modeling something external to and outside of human reason, that we have a relationship with that external thing, and not that we are just playing around in a world constructed by the mind.
An aside --
I'm still not sure I buy that the 'next point' or 'next irrational number' is a killer for getting away from the idea that the irrationals must be countable.
Let us say that I use this model instead: I program a random-number generator, which starts pairing the numbers 1, 2, 3... etc. each with a randomly generated irrational. If the random number generator pulls an irrational number it happens to have pulled before, it repeats until it pulls an irrational number not already paired.
In this way there's nothing in principle that should prevent us making the set countable. The only thing that made that seem impossible before was the trivial truth that there 'is no next point,' or rather, no 'next irrational number.' This method should avoid that issue entirely (which would restore our logical contradiction).
Note that this is still a potential and not an actual infinity in Aristotle's terms, by the way: it would take an infinite amount of time (and indeed, insofar as it is really generating and not designating pre-generated irrationals, it might take an infinite amount of time to generate the first one!).
I've got to be brief for the moment - short version is that the question is whether you can come up with a rule that uniquely pairs each positive integer with each members of the infinite set.
So, the set of all positive even numbers is infinite, but countable. Easy to see: Pair 1 with 2; pair 2 with 4; etc. etc. The set of all numbers divisible by 100 is likewise a countable infinity. Pair 1 with 100, pair 2 with 200, etc.
Now, by definition, a rational number can be expressed as a ratio of integers. 45/46, 10832/10833, etc. Off the top of my head, I can't tell you just how, but it's possible to pair these ratios with the actual positive integers - so that it's just as countable as the other sets.
With an irrational number, that's expressible only as an infinite, nonrepeating decimal. If you consider only the first 1000 digits of that decimal, you can pair it off with a positive integer. Problem is, the irrational number continues beyond that 1000th digit...in fact, there are infinite irrational numbers with the same first thousand digits.
By the bye, I woke up remembering Richard Feynman's autobiography, Surely You're Joking, Mr. Feynman!: Adventures of a Curious Character. Instead of telling his life story directly, he gives different anecdotes and experiences, jumping through the years if nothing interesting happens. It makes a good, quick read.
In one story, he talks about his morning chats as a grad-student with the mathematicians. They like to pose him conundrums. So one of them asks whether you can peel an orange so thin that the skin will surround the Earth - I am going from an old memory here. He replies "no." Then they ask, what about [mathematical theorem you've never heard of] that says you can slice a sphere so thin that you can make a wrapping of any size? (This is a memory from a 20+ year-old reading of one book, so I'm not getting it perfectly. You might enjoy looking it up given the things you've been discussing.) Feynman responded, "I assumed you were talking about a real orange." - the point being that, unlike a sphere, an orange is made of a finite number of cells or atoms, and cutting it the way that theorem proposed would go far beyond any meaningful concept of "slicing an orange." (You could spread the individual atoms out so far that they "surrounded" the Earth or the Milky Way, but it wouldn't be wrapping the way the math guys posed the problem.)
...it'll be this evening before I can go further, but I think you see the point. They're taking a mathematical object that is very useful in understanding the world (if you were considering the aerodynamics of an orange, you'd probably model it as a sphere; if you were considering its gravitatinoal pull, you'd probably model it as a point with the whole mass of the orange concentrated at that point). What the math boys were doing was taking the mathematical model further than the reality of the physical object would bear. Just the sort of thing that I'd describe as resonable back before we knew what stuff is made of - when the atomic theory could be, and was, denied by the smartest men of the age.
(The Stoic in Cicero's Nature of the Gods, putting down the very weak argumentum ad populum from the Epicurean, is rather vigorous on the subject - and in fact the crowning evidence for the atomic theory wasn't analyzed 'til the early part of the 20th century. But this - Einstein's paper on Brownian motion, which is what won him his Nobel, I believe - I think it is well known to you.)
Catch you later!
btw, the last I'd read, it wasn't actually proven that the number line is continuous - but I'd really have to check to see where that stands. A quick google put me here.
Joe:
The line logically has to be continuous for the irrational numbers to have the properties they have. They end up being unextended (which is necessary in order for a number line to have an infinite number of them between any two points).
I don't see why that creates an impossibility of pairing them aside from an actual impossibility -- if I go:
1 - pi
2 - a number somewhere after pi that begins 3.14159....(same as pi for as long as the program runs in the living universe)
3 - a number somewhere between the first two that begins 3.14159... (etc).
...we'll never get to a point that my 'kick back' rule comes into effect. But in theory these are three separate irrational numbers, which have been paired: we just can't actually run the program to a conclusion.
There is still also the priority issue with the continuum (which would hold if the number line weren't continuous as such, although I can't see how it could be otherwise). That means that we can assign meaningful names other than the infinite nonrepeating decimal that names to these numbers -- and in fact we do. When we name a number "pi" or "the square root of two," we're giving such a name by reference to the continuum. Since the continuum is prior, such a name is just as meaningful as the decimal name.
By the way, we have a similar non-contradiction issue without having to go as far as 'uncountable' sets. Pair the set of positive integers with the set of even positive integers. Logic tells us both that they must be the same size (because they pair up to infinity); but that, by definition, one of them ought to be half the size of the other.
Here, too, we're running into things that don't quite fit our model. They are, in human terms, irrational: and that's good!
(Hegel liked contradictions too, but for a very different reason.)
Pair the set of positive integers with the set of even positive integers. Logic tells us both that they must be the same size (because they pair up to infinity); but that, by definition, one of them ought to be half the size of the other.
No, no, no, no, no. The only thing that tells you "one ought to be half the size of the other" is intuition - not logic and not definition. This is because your intuition, like mine, is geared to finite rather than infinite quantities. (Which is also why it took Mankind so long to develop mathematics that dealt with the infinite.)
Now, while I was driving home, I came up with a way to pair each positive rational number uniquely with a positive integer (extending this to the negatives is trivial). It's probably not the simplest way but it's my own.
To understand this, you must understand that every integer, if it is not a prime number, can be uniquely factored into a set of primes. Thus 6 is 2 x 3 -- 27 is 3 x 3 x 3 -- 63 is 7 x 3 x 3, etc. The point is that for each of thse non-prime numbers, there is one and only one set of primes, multiplied together, that create that number. (Which is the basis of this xkcd.)
Okay, so express any positive rational number as a ratio of two positive integers, A and B. (Which, by definition, you can do; I assume you know that a repeating decimal can be expressed as such a ratio; I'm glad to show you how if you don't.)
Raise 2 to the power of A. Raise 3 to the power of B. That number is a positive integer. It is uniquely paired with the rational number A/B. (That is to say, there is no other rational number that, according to this formula, generates that positive integer.) Thus, 1/3 is paired with (2^1)(3^3)=54; 2/5 is paired with (2^2)(3^5) = 972, etc.
Thus, each rational number can be paired with a positive integer. And therefore the set of rational numbers is countable. This does not work with the irrational numbers.
I don't see why that creates an impossibility of pairing them aside from an actual impossibility -- if I go:
1 - pi
2 - a number somewhere after pi that begins 3.14159....(same as pi for as long as the program runs in the living universe)
3 - a number somewhere between the first two that begins 3.14159... (etc).
Because in between 1 and 2, the way you have defined it, there is an infinite number of other irrational numbers. Between 2 and 3, there is an infinite number of other irrational numbers. You can never, even in theory, even if "steady state" made a comeback and the universe was never going to end, come up with a rule to pair each integer with a unique irrational number, such that every irrational number is matched to a particular integer. You can do that with the rationals, as I showed in my last post. (Cantor showed this with more rigor than I have done - what were you expecting? I just made that one up.) Gamow's book, though old, has the best popular account I know, but it's been a long time since I read it (and even longer since I took number theory, on which I am drawing a little bit for these posts). The point is, Cantor's proof that the irrationals are not countable is the very basis for drawing that distinction, between countable and uncountable infinities (that is why the real numbers, which include the rationals and the irrationals, are uncountably infinite; so are the transcendent numbers, which are a subset of the irrationals; pi is one of these).
To return to what I was saying before - one thing that makes mathematics useful is that it helps us go into realms where our intuition cannot take us. These things are counterintuitive, and it is no knock on you that you have trouble with the distinction.
Because in between 1 and 2, the way you have defined it, there is an infinite number of other irrational numbers. Between 2 and 3, there is an infinite number of other irrational numbers.
Yes, but there are an infinite number of integers as well. You want to say -- well, but not as many as there are points! But of course there are; there are as many as you need. If you run out, just add one.
What I'm proposing is a kind of bracketing-fire rule. Think of it like mortar rounds. You've got a firm first coordinate, because it's defined by pi -- that is, not by the impossible-to-actually-generate decimal name, but by reference to the continuum that is prior to the point. Then fire one round to each side of it; and then start filling in the middle.
Each outermost point can be used to create new outermost points, and the middles filled in in the same way.
The objection you are raising is that the middles can never be 'filled in,' because each one has an infinite number of points. However, we have also an infinite number of rounds.
It's not a question of the universe never ending -- no one is (or ought to be) proposing that the rule could actually be followed through. You can't even actually follow through with a rule that pairs every integer with itself, if you have to actually do it: one-one, two-two, but the universe will still end before you reach the end.
And this, I think, returns me to a central point of this discussion - and a favorite theme of mine, the limits of intuition. And how far the questions, and reasoning, used in Greek physics can really take us.
Now, I believe that Aristotle himself, if he lived in our time, would readily embrace modern physics and all it implies, and modify his own views accordingly - but let's postulate a fellow named Asab (stands for "Aristotle's Smart-Aleck Brother").
According to modern physics, there is a smallest unit of time that can ever be measured - the Planck Interval. (It is mighty short.) We can, at least in theory, measure how far an object moves in a Planck Interval. We can't, even in theory, measure anything that happens in less time in that.
So old Asab comes along and says, "How far does an electron go in half a Planck Interval?" And I tell him, "There's no way ever to tell - we can't measure less than that." And Asab says, "Oh, ho, well my logic tells me what must happen in half that time..."
You see the problem? Asab's not just using "logic." He's using an idea about time that fits his own human experiences and his own human intuition - in which something that's steadily moving at 5 miles per hour for an hour, will move half that distance in half that time. But one of the big takeaways from my misspent youth as a failed quantum chemist is this: these intuitions are not useful guides once you get off the human scale. (Mathematics can take you a lot further than intuition in that case.)
I once daydreamed a world in which space is actually divided into tiny cells, and the whole universe consists only of "filled" and "unfilled" cells; when an object moves, one cell becomes filled and another unfilled, with no continuity. I have absolutely no evidence that any such thing is true. Yet it would be consistent with what we know, and make hash of Asab's "logic" (i.e., intuition) about how motion must work.
Another example, the Big Bang. Per the wiki, the physics that leads us to believe the Big Bang occurred can tell us what happened starting with the end of the Planck Epoch. - a period of time that, as far as we know, covers the first .0000000000000000000000000000000000000000001 second of the Universe's existence (assuming I counted 42 zeroes between the decimal point and the 1). With the physics we have currently got, we can't describe what was going on during the Planck Epoch. (Afterwards, the best answer we have is, a very rapid expansion of space, making it much less hot and dense.)
So along comes Asab, to ask, "What happened halfway through the Planck Epoch?" And I tell him, "We don't know - we don't have the physics for it yet." And he waves me away and says, "That's okay. Using my logic I will tell you what must have been happening." But his "logic" is working from intuitions about time and motion that don't apply here - and in truth he's got nothing to teach about that.
The only thing that tells you "one ought to be half the size of the other" is intuition - not logic and not definition.
I'm pretty sure it's not my intuition that's at work here. The definition of 'even number' is any number that can be divided by two. These occur every two integers. Thus, the set of all integers is twice as large as the set of even integers.
Yet, because we can pair them off, the two sets are also the same size.
The problem isn't that I find it counterintuitive. I don't, actually: I find it delightfully in accord with my intuitions about the nature of the universe. I expect rationality to break past certain limits: it's exactly what a neoplatonic thinker expects.
What you're concluding - or trying to conclude - is that there is no such thing as an uncountable infinity; that Georg Cantor had it wrong and that every infinity is countable.
Now as it happens there is a very elegant proof that by no means, including your "bracketing fire" idea, can you ever pair every irrational number with an integer, or create such a rule in the way Cantor meant it. This proof is the diagonal slash.
There is a very good account of it in Penroses's The Emperor's New Mind - I think Gamow's book also has it - but even the Wiki I linked to has an account that is better than anything I can come up with just now.
I'm pretty sure it's not my intuition that's at work here. The definition of 'even number' is any number that can be divided by two.
Yes...
These occur every two integers.
...and yes...
Thus, the set of all integers is twice as large as the set of even integers.
...and no! For a finite interval of integers that would indeed be correct. And because your intuition is geared to finite quantities, your intuition suggests that. But when you are dealing with infinite quantities, this is not so, nor has any reason to be so. This kind of mathematics is not irrational at all -- it is just very, very counterintuitive.
What I'm actually trying to conclude is that every infinity is the same size -- whether it is countable or uncountable is of no matter to me. That is, I'm concluding the very thing that modern physics appears to support. As cited above, "Real objects cannot have infinite charge or mass or whatever. But when scientists in the 1950s started calculating those quantities with their latest and fanciest theories, infinities kept sprouting up and ruining things. Rather than abandon the theories, though, a few persistent scientists realized that they could do away with the infinities through mathematical prestidigitation. (Basically, they started calculating with and canceling out infinity like a regular old number, normally a big no-no.)"
The problem you're having in convincing me, I think, is that you haven't actually worked through the Physics. :)
You keep trying to tell me that Aristotle would agree with you if only he'd had the benefit of your education, but perhaps there remain a few things to learn from the old master. All of these thought experiments you're citing are there -- the problem re: Plank length is a Book VIII problem, although obviously it's phrased in different terms. It's also a problem about the atomists, quite familiar to Aristotelian and neoplatonic thought. Your contention from a recent discussion that space and time can't exist separately, and so that there is no 'before' to talk about is a Book VII issue, in which Aristotle comes to the same conclusion that you want to come to as well.
The problem about irrational numbers being infinite is a problem about the nature of a continuum, which Aristotle discusses at very great length in the Physics.
There should be no conceptual problem in thinking of infinities as a kind of constant like the speed of light. Consider the famous schoolhouse problem of two cars driving towards each other, versus two starships roaring at each other at substantial percentages of c: in the one case you add the velocities together to get the speed with which they are closing, and in the other you need a whole different approach.
That seems to be what physicists are doing by canceling infinities, and it seems to work. If the model says that can't be done, we just haven't figured out the right way of doing it; and you should join me in trying.
Now that seems to me what Aristotle would say: it's empirical, in just the way that you (and he) like.
But let's back up. Here's a bit from the original post.
There is another problem, though, which is that we can also obtain knowledge through contemplation alone: for example, we can come to knowledge of mathematical truths simply by thinking.
The Cantor proof you cited a moment ago is an example of this. The main thrust of our discussion has been against it -- in favor of needing 'to go back to the world' to learn about things like c behaving differently.
Since we're now 55 posts into this discussion, where do you think we stand on the question of whether we can attain mathematical truths by contemplation alone?
What I'm actually trying to conclude is that every infinity is the same size -- whether it is countable or uncountable is of no matter to me.
The whole point of "countable" versus "uncountable" infinities is that this is not so - that is why Cantor is remembered at all. To do this, he had to come up with a concept for "size" that applies to infinities - which is a matter of pairing them off with mathematical sets, not as a final result, but as a process.
After a good night's sleep, I woke up remembering just how to explain the diagonal slash in terms of your example - an interval bounded by pi.
Imagine your numbers set out in grand array, paired off with the positive integers, like this:
1 -- 3.1415926535....
2 -- 3.1415950112....
3 -- 3.1415926536....
...and so on, so that every irrational number you're creating is paired off with an integer, by means of "bracketing fire" or whatever.
Now, if I can create an irrational number that is not in your array, no matter how big your array gets, then you are not able to pair off all the irrationals in that interval with the positive integers.
And I can do that. Call my new number X. The first six digits are the same as pi - 3.14159... The seventh digit is going to be a three (I added 1 to the 2 that occurs there in pi)
For the eighth digit, take the eighth digit of the number you have paired with integer 2. Change that number to something else. That is the eighth digit of X.
For the ninth digit, take the ninth digit of the number you have paired with integer 3. Change that digit to something else. That is the ninth digit of X. Keep going.
Now, X is going to lie inside your interval, be a nonrepeating decimal, an irrational number. Yet it is different from every number you've got paired with an integer in the interval. Thus, they can't all be paired, not only not as a "final result" (which is by definition impossible), but not even as a process that takes you out to infinity. This works no matter how tightly you draw the interval.
The rational numbers, as I showed you above, can be paired with the integers...not as a final result, but as a process that takes you out to infinity.
I'm told Cantor actually had a different proof to start with, and developed this later. But this is the one that makes it into the pop-math books, because it's much easier for the readers to follow, and is beautiful in its elegance.
But the real point is this - you were trying to say that mathematical results in this area are irrational -- and I am trying to show you that they are not irrational, even though they are counterintuitive, and understandably take an effort to grasp.
The rest will have to wait 'til lunch break or nightfall.
I read the proof, and your explanation of how you apply it, but I still disagree. We'll get to X, because we're firing an infinite number of rounds until we've hit every point between 2 and 3. It isn't important that we get it on the first stroke, because of course there will be an infinite number of X-type points that we miss to start with. The point is that the rule continues to fill in the gaps forever -- so it's OK if the gaps are infinitely big.
The other point is that we calculate the place of the irrational not by its name, but by reference to the continuum -- thus, starting with pi -- so that we have a reliable way of bracketing. We're going to be shooting infinitely close to our first coordinate (and every other coordinate), so we need a coordinate that is precise and reliable.
Actually, I was thinking a bit more about this, and it occurs to me that there's a significant difference between a line with points, and a number line with numbers. All the points are of the same type, but some (although vanishingly few, as it turns out) of the numbers will be rational.
So we'll need to modify the rule to ensure that any pairing that finds a rational number is re-issued, because it won't do to pair a rational number.
Now, pause and consider that the intuitive problem may be on your side. A countable infinity is supposed to be a different size than an uncountable one because it is a subset: the set of real numbers is uncountable, but the set of rational numbers is both countable and a subset. Our diagonal proof is elegant because it shows you a proof that there will be elements not included, etc.
That's the same kind of argument about the even set and all sets. It's not, in fact, 'intuitive' in the sense I think you are using the word, but rather a logical argument that shows that the proof follows from the definitions. We've given definitions of sets and subsets and inclusion; and we've given a rule for creating an infinite set S0, and then S(m,n) pairings. In each case, the argument follows from the definition.
Thus, your arguments are completely rational -- but the physics seems to show that they aren't true. Infinities do seem to work more like c: they serve as a ceiling, beyond which what ought to be true isn't true. It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not rational given the system, but it is true.
So maybe you'll like it better if we say the world isn't rational, rather than that math isn't; but if math is going to map to the world, it's going to have to know when to bend or change its rules. What follows from reason ceases to apply.
That last thing you said isn't so at all. To answer you best, let's go back to this --
Since we're now 55 posts into this discussion, where do you think we stand on the question of whether we can attain mathematical truths by contemplation alone?
Now, I thought it was physical truths we were talking about, derived through mathematics, but this is how it seems to me --
#1, mathematics, the kind we can apply to the physical world, starts with observable facts in the real world. (Objects add, multiply, and divide the way numbers do.) Observable reality gives us postulates we can use to derive more mathematical facts. The positive integers, the axioms of Euclid - these came about in this way. And the calculations we make using these postulates - correspond to the real world.
#2 - Sometimes, however, our ideas about the world that lead to mathematics, whether based on intuition or observation, lead to results that do not fit the world. In that case, we have to change our assumptions in order to get math that works out physically.
So, if we reject the idea that an infinite series can sum to a finite sum -- we get Zeno's paradoxes, which do not fit the world we observe. If we accept the idea, however counterintuitive it may be, and develop a mathematics accordingly, we get results that are astoundingly accurate. I gave you the example of the Central Limit Theorem before (which does require calculus and much more). Another famous example is non-Euclidian geometry - geometry that begins by changing the fifth axiom of Euclid; and as you probably know, fits elegeantly with general relativity, and is actually useful for relativistic calculations.
So mathematical proofs are indeed an exercise in logic - but if the math is going to be fruitful, and the "truths" are going to correspond to something in the outside world, the assumptions from which we begin have got to be based in that world. And when the results don't fit the world, sometimes those assumptions have got to be modified. The reward is that mathematics takes us places where "raw" intuition never would -- in ways that fit the evidence.
"Thus, your arguments are completely rational -- but the physics seems to show that they aren't true. Infinities do seem to work more like c: they serve as a ceiling, beyond which what ought to be true isn't true. It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not rational given the system, but it is true."
But didn't we just determine that there are two different types of infinities, therefore, why would we assume that all infinities should follow the same rule? Also, you say it's not rational, based on our interaction with the universe, but we can't interact with things moving at hyper velocities for instance, and apparently physicists are making the calculations work if they cancel infinity- so apparently we do have some kind of rational handle on infinity. Now, it may turn out it's wrong later, like Newtonian physics, but for a while, we thought that was bulletproof. In the meantime, I'm still not persuaded we've negated the Kantian model (though I don't really believe it). So long as we can develop rationale for these ideas under which what we predict and what we observe are consistent, and we seem to be doing that, Kant could be right. I don't think it hinges on executability- only on the rationale and the evidence matching up, right?
A countable infinity is supposed to be a different size than an uncountable one because it is a subset...
No, sir, that's your idea, not Cantor's. A countable infinity is supposed to be a different "size" than an uncountable one -- and note that Cantor has expanded the concept of "size" to account for these different kinds of infinities -- because the one can be matched, by rule, to the set of integers and the other can't. The set of positive even integers is a subset of the positive integers, but they are both countable infinities and are the same size -- as a matter of definition.
Your "shooting" analogy is an intuitive one, the kind of thing that's very useful for understanding problems on the human scale. It doesn't work so well for infinities. As I mentioned above, comparing infinities is a matter of comparing processes - do you have a set of numbers to which the positive integers can "catch up" by a process you can define? If yes, it's countable; if not, no. The set you've defined is uncountable, because if you pair it with the natural numbers, by any means whatsoever, it never catches up - an X can always be defined that your process won't reach.
This is not the case wiht the rational numbers; the positive integers can "catch up," and so it is a countable set. That's the best metaphor I can think of - but going with metaphors and intuitive analogies gets you only so far in mathematics or modern physics, and then you are here.
Our diagonal proof is elegant because it shows you a proof that there will be elements not included, etc.
Right...
That's the same kind of argument about the even set and all sets. It's not, in fact, 'intuitive' in the sense I think you are using the word, but rather a logical argument that shows that the proof follows from the definitions.
Exactly.
Thus, your arguments are completely rational -- but the physics seems to show that they aren't true.
Now you're mixing up a couple of different things -- special relativity, which is what you're talking about, has nothing to do with the difference between countable and uncountable infinities - not as far as I know.
Off the top of my head, in fact, I didn't know any physics that deals with that distinction - I did a quick Google and found an abstract or two I did not understand; so I think there may be some, but these simple problems in special relativity you're describing aren't among them.
It should be true that two spaceships traveling directly at each other at .75c each are closing at a rate of 1.5c, but in fact it won't even be 1c. That's not rational given the system, but it is true.
Incorrect. In this case, it's not even a matter of developing a new kind of mathematics, but of changing some ideas about the physics, based on the experimental data. (Special relativity is derived using ordinary algebra and calculus; it's general relativity that requires the non-Euclidian geometry. Two spaceships approaching each other at uniform speeds require only special relativity to describe their motion.) The physical assumption that underlies it is - if spaceship A is firing a light beam at spaceship B, the light beam is travelling at speed c away from spaceship A. It is also travelling at speed c towards spaceship B; its velocity, relative to both these moving objects, is exactly the same. Which is a deeply strange thing, but if you change to this assumption, you get results that fit the world very well indeed (one result is that nothing will go faster than light relative to anything else; which is the result you are talking about).
There is nothing irrational anywhere in the process, just as there is no "flagrant violation of the law of non-contradiction" in Cantor's way of comparing infinities, a separate topic.
One of the things that Aristotle often says in the Physics is, "Let us make a fresh start."
There is a reason why it is impossible to divide a line both at a given point and the point next to that first point. The reason is that 'the point next to it' is a phrase that does not refer to any extant object. The reason is not that you cannot divide at an infinite number of points, or indeed at any other point that does exist; it's just that the concept "next to it" doesn't fit.
Thus, let's say I wish to assign a round to every irrational 'point' on the line that composes the set of real numbers (we shall call this set "R"). I place an order for R rounds.
But I don't actually need this many: I only want to hit the irrational points, so the rational ones ("r") will be left out. I therefore should only need R-r rounds to hit all the points I want. Since I ordered R rounds, I'll have extra!
Or possibly not; since the set of irrationals is an uncountable infinity just like R, the fact that I'm subtracting a countable infinity may leave me with just enough. So, I will have either more than I need or exactly as many as I need.
But we can further specify the relation. The difference between the countable infinity r and the uncountable infinity of the irrationals is also able to be specified: between each member of r and the next member of r, there is an infinite number of irrationals. Thus, the irrational set is equal to r multiplied by infinity.
Now my claim amounts to this: squaring infinity doesn't make any more difference than multiplying it by 1/2. The set "r" is infinite by nature; thus, if you take the square root of r and R, you'll end up not with two sets of different 'sizes,' but with something like "1=1".
As for special relativity, I include the analogy only as an analogy: I mean to say not that there is a relationship between c and infinity, but that we can recall the problems of c in relativity theory to remind us that we sometimes come up with unexpected limits. I think this is one area where very elegant mathematics are hitting a metaphysical limit; I believe I understand Cantor's argument, but it strikes me as fundamentally the same as the claim about evens and all integers. It's clear enough, looking at the number line (or the diagonal line); but it doesn't seem to be true.
Or rather, it does and it doesn't seem to be true: in the same way that I expect for my supply of R rounds to be both more than enough, and exactly the right amount.
By the way:
Many philosophers would have absolute fits at the idea that the law of noncontradiction can be violated (let alone is being violated, ever at all; and especially is violated by something as rational as mathematics). They'd want to join you in finding some sort of distinction that makes it all rational and orderly.
There are also Catholic philosophers and theologians who defend it to the point of saying that even God cannot do something that violates that particular law, or other laws of logic. This is often floated as a defense against atheistic arguments like, "Can God make a boulder so heavy he cannot lift it?" Either yes or no must be true; either claim is incompatible with Catholic claims about God. So the argument that God is ruled by the laws of logic is a way of avoiding that kind of anti-theistic argument.
Neoplatonic philosophy differs in that it believes that at the level of nous, noncontradiction breaks down. Actual infinites (as opposed to potential infinites) do not exist outside the realm of the intellect; therefore, it isn't important that noncontradiction should apply. It can be true both that there is a 1/2 ratio of all integers to even integers, and that this truth doesn't apply to the set. It can be true that we'll have exactly enough rounds, and more than we need. This is nonproblematic, because at the noetic level contradictories coexist in a single unextended whole.
It also occurs to me that the irrationals are going to be the same set of numbers between each integer, which offers interesting potential: maybe we can triple-pair them, so that you'd have one integer to denote which integer provided the field, and the second to denote the specific irrational being paired. Pi could be 3.0 -- that is, 3 to indicate it was in the space between 3 and 4; and 0 to indicate it was the first paired irrational. We'd need a table after that, since you can't go directly from Pi to 'the next smallest' or 'next largest,' but in theory once you map the set for one integer range, you've mapped it for all of them.
Now, of course, this can't actually be done: but it leads me to think it might be a way of squaring the idea of the sizes of the two infinities being the same. In this way we can see that there's really only one irrational set; it just happens to recur between each integer.
But I don't actually need this many: I only want to hit the irrational points, so the rational ones ("r") will be left out. I therefore should only need R-r rounds to hit all the points I want. Since I ordered R rounds, I'll have extra!
R and r, as you have defined them, are sets, not numbers. When you treat them as numbers, and say "I need only this many rounds," you are quite misusing the concept, and your statement doesn't make sense.
Now, if your intent was to subtract one infinity from another, that doesn't make sense either - infinities are not numbers that can be manipulated the way 6 and pi can.
If your intent is to take the real numbers as the unity of the rational and the irrational, well and good. You can certainly consider a subset of an infinite set and -- this is the key point -- infinite sets can have infinite subsets as well as finite subsets. Thus, "the set of all even numbers" is a countably infinite set; "the set of all numbers divisible by 100" is a subset of this set; but both are countably infinite. And in the sense that Cantor used the term, neither of them is "smaller" than the other, even though one is a subset of the other.
In any caes, R and r and the other objects we're talking about are still sets, and they are not numbers. Discrete objects, like "rounds," can be paired with the positive integers and are therefore, at most, countably infinite.
Now my claim amounts to this: squaring infinity doesn't make any more difference than multiplying it by 1/2.
That's right. Neither one of these operations makes any sense at all, no more than dividing by zero does. Dividing two by zero doesn't give you a number twice as big as dividing one by zero. Both are undefined and neither makes any sense. "Taking the square root" of any infinity, that makes no sense either.
(The way you work with infinities in applied mathematics is by using a variable, and taking the limit as that variable goes to infinity. Now if you were to take an formula like 2x/y, and take the limit as x and y both go to infinity, the limit is obviously 2, not because one "infinity" is bigger than another or because you're actually "dividing infinity by infinity," but because in the process of going to the limit, the expression on top stays twice the size of the expression on the bottom.)
Many philosophers would have absolute fits at the idea that the law of noncontradiction can be violated (let alone is being violated, ever at all; and especially is violated by something as rational as mathematics). They'd want to join you in finding some sort of distinction that makes it all rational and orderly.
But I don't need to "find some sort of distinction" - it already is rational. (I'm not sure what you mean by "orderly" in this context.) It is sometimes counterintuitive, but I have never yet seen advanced mathematics that is irrational. You've asserted once or twice that it is, and violates non-contradiction, but you haven't shown how...or rather, your arguments that it does break down once you look at the true definitions we're working with.
Interestingly, this puts you in company with a philosopher who doesn't get much time from the academics - Ayn Rand. The Introduction to Objectivist Epistemology includes the transcript of a little seminar with her and some academics, where she declares that "mathematics after Bertrand Russell" is a fundamentally irrational enterprise. Which I'm afraid showed only that she did not understand modern mathematics, and some of his pop-quotes about "the science where we never know what we are talking about" had quite bamboozled her. (Interestingly, she liked to frame the battle of good and evil on the earth as a battle between Aristotle and Kant - Kant she blamed for the worst excesses of "we can't know anything" indecisiveness - see her West Point address for an example.)
There are also Catholic philosophers and theologians who defend it to the point of saying that even God cannot do something that violates that particular law, or other laws of logic. This is often floated as a defense against atheistic arguments like, "Can God make a boulder so heavy he cannot lift it?" Either yes or no must be true; either claim is incompatible with Catholic claims about God. So the argument that God is ruled by the laws of logic is a way of avoiding that kind of anti-theistic argument.
Yep, I once heard William F. Buckley and Jerry Falwell agree that, not only can God not lie, he can't sin either. And since, to all appearances, God is an imaginary construct, they can give him whatever attributes they like. The difference between what they were doing, and what I am doign here, is that I do know something tangible about the mathematics we're discussing, have done some of it, used some of it, and worked through some of the proofs. (Sometimes long ago - which was why I was a little slow to remember just how the diagonal slash worked, for example.) So I am not simply repeating the dogmas of a Church, or demanding that an invisible, undetectable Deity conform to these, but looking at what men have actually done. And in doing so, encouraging you to stay out of this territory. (Three xckd's in one thread! That cartoonist had truly added something to this world.) Were I wrong, who knows, you might be able to show something irrational or contradictory in the mathematics we're discussing. You just haven't done so yet.
I've been waiting for two days for you to cite that last xckd. :)
It's true, in this case, that I don't have the background to do more than punch at you and see what you come up with. My background in symbolic logic is pretty good; in math general, it's largely limited to a fairly complete understanding of the odds of poker.
We're in the place where we are because you haven't done the Physics, and I haven't done the math. You keep raising arguments that you haven't thought through to the same degree that Aristotle did; and I don't really understand the mathematical concepts or terms, as you rightly point out from time to time.
I'm not sure we have enough common ground for me to explain to you why your arguments aren't going to shake out, or for you to explain to me why my attempts to explain them to you in mathematical terms aren't convincing.
Not yet. But if we keep at it for another few years, who knows? Maybe we'll learn something. (At least you've probably learned that telling a philosopher that something is impossible is likely to produce an attempt to do the impossible thing; and I'm still not convinced we can't sort out a good way, once we make up our minds and sort out our terms.)
...not only can God not lie...
That's funny, because I just attended a colloquium on Medieval Jewish and Islamic philosophy in which the translator (who could read Hebrew, Arabic and Greek) discussed Maimonides' praise for God's capacity for the ruse. I'm not sure that goes as far as a lie, but it's an interesting concept.
It is (and I think that some Muslims describe God as the "ultiimate deceiver" as well).
At least you've probably learned that telling a philosopher that something is impossible is likely to produce an attempt to do the impossible thing...
No doubt. Depending on just what it is, many scientists are similar. Sometimes, though, the question is just nonsensical. A quote from my freshman chemistry textbook --
"People sometimes ask me whether there might be elements other than those in the Periodic Table. I tell them that this is like wondering whether there could be another whole number between 6 and 7. Unfortunately, some people think that is a good question, too..."
...and we leave aside the poor cranks, neither scientist nor philosopher, who are told that they can't square the circle or trisect the angle by the rules of Greek geometry, and waste lives and paper trying to prove they've done it.
(In that case, thanks to analytic geometry and more besides, the proofs of impossibility are out there.)
As for Rand, I can't say I've ever found her interesting enough to read. I still don't; I got about two or three paragraphs into her piece, then started trying to find the part where she talks about Kant; discovered she's running Kantians and Hegelians together; well, in any case, Kant wasn't a bad guy at all. He was a hypochondriac, who set up a very rational system for himself to avoid admitting disorder into his life. He meant well, but his order is so orderly that even Kantians normally shy from it (especially when he turns, as he does now and then, to the subjects of capital punishment or breast-feeding).
You know, one thing that Rand is particularly wrong about in this piece is her assertion that the Metaphysics is "the basic book" of philosophy.
In fact, students traditionally weren't allowed to read it until they had mastered the Physics, the logical texts (in the Middle Ages known as the 'Organon'), the works on the natural sciences, and a few other works like De Anima.
Until the explosion of knowledge in the modern period, it was still possible for an individual to know everything that we knew as a species. That prevented talking-past-each-other discussions like this one, where backgrounds and languages are too different for a meaningful exchange of ideas; but at the cost of losing all that knowledge.
It's a good argument for a better replacement for humanity -- or at least an upgrade for our minds and lifespans. We really can't do it anymore. Most of those I've studied with have at a pair of Ph.D.'s in order to keep up with some bare minimum: I know a philosopher of science who has Ph.D.'s in physics (focused on relativity) and metaphysics; another with degrees in math and philosophy of math; I have only a master's in history, to go with the doctorate in philosophy I'm after. Someday I hope to round it out as well.
But even then, there's so much now that there isn't time to learn -- even if you don't take time off for wars, or families, or the joys of poker.
Too right. May we live long enough to see that start to happen!
P.S. - And I have noticed that any trained academic philosopher who reads Ayn Rand seems to agree that she did a terrible job describing the work of other writers, especially Kant. (And her epistemology relied heavily on psychological ideas, such as the tabula rasa that are simply not true; that spoils her attack on public education, more's the pity).
It's a shame she took the path she did, because she really was a fantastic expositor of some very important ideas. Her essay, "The Anti-Industrial Revolution" - title essay in her book, The New Left: The Anti-Industrial Revolution, is an extremely good attack on "green" thought, and its appalling moral standing. (I read several books of her essays; I think that was the best one. On a good day, on the attack, she was truly unmatched.)
In fairness (as you know, having looked at him recently) Kant is not an easy philosopher to understand. He's one of a few philosophers that there are competing schools about, too, where some think he meant to say X and others Y.
For example, I belong to the school that interprets his moral theory less on the more-famous Groundwork to the Metaphysics of Morals, and more on the later Metaphysics of Morals -- but it's much less well known, though for the most part it's much more obviously sensible given the moral positions you'd expect from a European gentleman of cosmopolitan leanings in his time.
However, I think it is fair to say that he is more popularly read (based on the Groundwork) as offering a program to radically rethink moral philosophy. I think he was just trying to make a secular argument for the ordinary moral beliefs of his day; but many like to take his Groundwork concepts about how to formalize morality and run with them into interesting and sometimes radical places.
Rand may have had one of those teachers! But it's hard to see how she confuses his teachings with Hegel in any case. (Except that they're both German, and both take a huge amount of work to decipher! I assume she was against Hegel because Hegel was such a crucial figure for Marx.)
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