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April 29, 2005
Mothers of Invention
I had thought that Putnam, Harman and Kripke had the imaginative-counterexamples-to-necessary-truths market sewn up, but I have just learned that I was wrong. You might remember that Putnam, for example, argued that "all cats are animals" expresses a contingent truth by describing a situation in which we would say the sentence is false: if it turned out that all the things which we call "cats" were (extremely sophisticated and well-disguised) robot spies from Mars, then we would say that "all cats are animals" is false.
Harman described a similarly unlikely situation that would make the sentence "red is a colour" false, and though I forget the exact description and I don't have the book handy, this is roughly it: if we discovered that we have the visual experience of redness in response to a certain frequency of sound emitted by an object, rather than in response to a certain wavelength of light, then we would say that "red is a colour" was false.
Kripke turns a similar trick for "Gold is a yellow metal" (which was one of Kant's examples) and there's a good chance that I'll be posting more about what these kinds of examples show, since I have quite a bit to say about it.
These examples are sometimes - though not always - used in defence of radical empiricisms, where by radical empiricism I mean versions of empiricism which reject any kind non-a posteriori knowledge, even that based in meaning. One criticism of such views is that they cannot explain de dicto necessities, such as "2+2=4" or "triangles have three sides" or "all cats are animals;" suppose we allow that experience may somehow bring us to believe such claims, how could it ever bring us to believe that the claims are necessary?
One response open to the radical empiricist is to say that the claims are not necessary; we only think that they are because we are not imaginative enough to come up with the possible situations which would make them false.
In the history and philosophy of mathematics, something like the radical empiricist view is usually associated with John Stuart Mill, and so I should not have been surprised to learn from Coffa's The Semantic Tradition from Kant to Carnap (though I was a little surprised - I had mistakenly thought these hyper-imaginative counterexamples were a relatively recent phenomenon) that Mill also ran the no-imagination-crazy-counterexample defence.
In arithmetic, for example, our commitment to the law that 2+2=4 would vanish if whenever two pairs of things "are placed in proximity or are contemplated together, a fifth thing is immediately created and brought within the contemplation of the mind engaged in putting two and two together." The production of this fifth thing must be "instantaneous in the very act of seeing, [s]o that we never should see four thing by themselves as four: the fifth thing would be inseparably involved in the act of perception by which we would ascertain the sum of the two pairs."...Clearly Mill was thinking about adding up things like rabbits or cows, not things like solutions of third degree equations or Roman consuls. As Frege would point out in the Grundlagen (1884, secs 7 and 8), the later are not easily "placed in proximity" or involved in "acts of perception." A world in which, when someone adds the first two Roman consuls to the next two a fifth one appears, presumably with his distinct proper name, his own political record, and so on, is not a world at all, but the product of a confused mind; for in that world the decision to add would alter the past, and on pain of contradiction there could not be one person adding a group of objects and another not.(47-48 of Coffa)
Coffa's choice of rabbits, and then cows, as examples of things which Mill must have had in mind confused me on a first reading, since these are creatures capable of reproduction, and in fact rabbits are famous for breeding like...rabbits. Solutions to equations and Roman consuls cannot, so the examples might suggest that the crucial thing is that certain objects, when left together long enough, can produce more objects. But I think Coffa recognises that Mill's counterexample is even more imaginative than this.
Sometimes when one watches television, channels which feel bound to protect their viewers or subjects will cover a part of the image with a solid black rectangle. Sometimes the rectangle covers a suspect's eyes, in an effort to protect the suspect's anonymity, and sometimes it covers a subject's genitalia, in an effort (presumably) to preserve the delicate flower of our viewing innocence. Now suppose your eyes and perceptual system did this kind of thing automatically (I can't help wondering whether there are mental disorders like this.) And now, to take things a step further, suppose that instead of covering up something with a rectangle, your perceptual system instead creates an extra object whenever two pairs of objects are brought together. For example, suppose you look to your left and see two apples, then you look over to your right, and see two more apples. Then you close your eyes, reach out to the right and grab the apples that are over there. You bring them over to the apples on the left and then open your eyes and look left. What you see is the four original apples, plus a fifth - the product of your perceptual system under these kinds of circumstances. This, I'm guessing, is the kind of thing Mill is getting at with his:
[The production of this fifth thing must be] instantaneous in the very act of seeing, [s]o that we never should see four thing by themselves as four: the fifth thing would be inseparably involved in the act of perception by which we would ascertain the sum of the two pairs."
It is very unclear to me that Mill's story can be told consistently. Whilst I think the non-standard story about the apples is perfectly clear, and conceivable, Mill takes it that this story can be generalised, so that "we never see four things by themselves as four." But suppose I see two (sterilised) rabbits. How many rabbit ears do I see? Mill's answer should be five, but, I wonder, where is the extra ear attached? Is it just floating in space near the whole rabbits? Perhaps Mill would tranquilly absorb this bizarre consequence - it's not as if he is trying to persuade us this is actually what happens with our perceptual systems. But suppose I am counting attached rabbit ears? Then perhaps Mill could say the extra rabbit ear is attached in an abnormal place. But suppose I am counting rabbit ears attached in the normal place? Can we fit two ears in one normal place? And here's the clincher, I think, suppose I want to consider both how many ears attached to normal-looking rabbits there are and how many normal looking rabbits there are. If there are two normal looking rabbits there, then, on Mill's assumptions about our perceptual systems, what we should see is exactly 5 ears-attached-to-normal-looking-rabbits and two normal-looking rabbits. But that is surely something that it is not possible for our perceptual systems to represent.
Mill's attempt at a counterexample to the claim that "2+2=4" expresses a necessary truth reminded me of my old German linguistics teacher Chris Beedham, who once tried to convince me, not just that "1+1=2" was contingent, but that was not really true, since if you add one raindrop to another raindrop, the result is one larger raindrop, not two. I think my response at the time was to say that he should not think of addition as bringing things together (and perhaps doing a little pushing to overcome the surface tension.) And though I wasn't quite sure of the correct positive response, I was fairly sure (and am still sure) about the negative part: this example doesn't show that 1+1 is not 2, and the problem with the example is something to do with the misinterpretation of the addition symbol.
People are always trying to talk me out of believing the basic truths of arithmetic. On a Greyhound bus two years ago I met an "FBI interrogator" who argued that "our math" was only true for us, since a different culture might interpret "2" as meaning 5 - in which case "2+2=4" would be false and "2+2=10" would be true. I imagine - though I do not recall - that he was a little less careful about the use/mention distinction that I have been in the retelling of this argument. And my grounds for thinking this is that the argument as a whole is based on a confusion between a sentence and its content: yes, relative to some other language, "2" might refer to 5, and that would make the sentence "2+2=4" false (with respect to that language), but does not show that what the sentence says (namely that 2+2=4) could be false, since in that language "2+2=4" does not say that 2+2=4, but that 5+5=4. This didn't convince the FBI interrogator, but then, he wasn't really listening and we moved on to talking about whether it was ok to "get a bit rough" with suspects if the crime they were charged with was especially horrible...he was wrong about that too. Greyhound - it's the new Clapham omnibus.
Posted by logican at April 29, 2005 01:18 AM
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Comments
Jon: But addition is disjoint union.
Beedham: You can't go changing the rules of the game there, sonny jim, I am adding the drops together. Retort!
Jon: Very well, you have not defined what a drop of water is. What, amount, pray tell, is a drop?
Beedham: Oh, you know what a drop is. About 1 ml. Oh very well, let's just call it 1ml.
Jon: So what volume of water do you have if you "add" two drops of water together?
Beedham: Are you daft? 2 ml of course!
Jon: So...?
Beedham: Oh dear, I was wrong.
Posted by: Jon at May 3, 2005 02:42 AM
Pseudo-Dr Beedham: What rot! A drop of water need not contain a set amount of water. I don't have to get out any finely callibrated measuring equipment to know that drops of water are falling from the sky. The amount of water in a drop can vary. In my example, we have two drops before we put them together, one drop afterwards. What could be simpler?
Posted by: Gillian Russell at May 3, 2005 02:54 AM
Jon: But then what happens when you iterate the process? You'd be calling a flood a drop! Unless you place precise bounds on the size of a drop, you will run afoul of this problem. You are playing on the ambiguity of the measurement - your example would not work were we to switch to cups instead of drops.
Pseudo Doc B: Humph! You just don't get it do you my boy? Two drops to one. S-i-m-p-l-e. I never said anything about more drops or cups!
Jon: That's just it, addition is defined recursively. If it holds for 2 drops, it must hold for any number of drops. Besides, if we define drops physically, then their size does not vary as much as you claim. They grow just large enough so as to overcome the surface tension holding them in place (to a tap, say) and subsequently plummet to their eventual demise. An amount of water almost twice the size of one of these drops would not decide to band together and cling on for dear life!
Posted by: Jon at May 3, 2005 04:01 AM
your example would not work were we to switch to cups instead of drops.
Psuedo-Dr Beedham: Let's be clear on what I need to make my point. You claim that 1+1=2, and by this you mean something quite general, namely ALWAYS AND EVERYWHERE, WITH RESPECT TO ANY SORTAL AND ANY POSSIBLE SITUATION, one thing added to another makes 2. That is, no matter what time it is, or where we are, or what kind of thing we're counting in what imaginary situation, it's still the case that 1+1=2. Right?
Gillian: sounds right to me, Pseudo-Dr Beedham.
Pseudo-Dr Beedham: And now let's be clear about what would be required to refute this claim. All I need is a single example of a case where adding one thing to another does not give us two things. Just one tiny example would make the general claim false.
Gillian: Certainly, indeed, you are equal to Zeus in your mind's resource.
And my example is the one involving raindrops: add one to another and - at least sometimes, - they join together to form a single drop. That is my counterexample, and it really isn't relevant whether it would work with cups or larger drops or if we put detergent in the water to reduce the surface tension.
(Jon: didn't I say that?)
Gillian: I agree that these other cases are irrelevant, but did you notice the other part of Jon's argument:
That's just it, addition is defined recursively. If it holds for 2 drops, it must hold for any number of drops.
Pseudo-Dr Beedham: But of course, nothing escapes me. I see that he is appealing to the standard recursive definition of the addition function. For the natural numbers it looks like this:
(0+x)=0
S(n)+x = (n+x)+1
I was tempted to say that this - though endorsed by the blinkered technicians of your hegemonic mathematics - is inadequate as a definition of addition, but I've just realised that it is perfectly consistent with my claim. My example shows, not merely that 1+1=1 is sometimes true, but that the successor of 1 is, sometimes, ... 1. The second claus of your definition implies that the successor of 1 (namely, 1), when added to 1, is equal to 1+1 (that is, 1) added to 1. I agree with this, even though I think 1+1=1 in this case, since both sides of the equation are still equal (just to 1.) Thus the brilliance of my eclectic mathematical vision can encompass the dogma of your unimaginative and stilted academic vision, yet goes beyond it...! (drops into a dead faint)
Posted by: Gillian Russell at May 3, 2005 05:37 PM
Jon: Once more you are playing at the vagaries and ambiguities of the English language (though an otherwise fine and robust language it is). For a mathematician, "addition" means something very specific. In the context of bringing together collections, it is disjoint union. However, if we use the word outside of an appropriately specific context, it is easy to obtain "counterexamples" by merely abusing the word. Heck, here is one for multiplication: If one rabbit multiplies with another rabbit to form 6 baby rabbits, then we have 1 x 1 = 6! This is, of course, just another abuse of our own fuzzy interpretation of the word "multiply". It is not a counterexample, it is a confusion on our part - we have parsed "multiply" in two different ways and so too for your "add" example. It is really not essentially different from saying "you and I make one" and trying to derive 1 + 1 = 1 from that.
Pseudo Doc B: Hmph! Humph! I shall dream up a riposte to that soon enough (most likely to coincide with Gillian's next update!).
(Gillian: yes you did mention surface tension - I just put it in the context of the formation of the drops)
Posted by: Jon at May 3, 2005 06:46 PM
I hate to interrupt the flow of the dialogue, but the counterexample reminds me of a related problem one of my students had with the platonist view of mathematics the other day. When the sentence "2+2=4" was on the board, he asked whether there was supposed to be just one abstract object 2 in platonic heaven or more of them, and if there was just one, then how can we add it to itself? I had to point out that even if you consider "1+3=4" then you should notice that 4 isn't itself an entity composed of two parts that you get by pushing the 1 and the 3 close to each other - this is just a three-place relation of a certain type, and two of those places can be filled with the same entity, as in "2+2=4".
Posted by: Kenny Easwaran at May 3, 2005 08:58 PM
S'ok, I think Jon's right that the mistake in the raindrop counterexample is one of confusing addition with something else, such as "bringing close in space" and so my inner Dr Beedham has fallen strangely silent. Anyone who wishes to is welcome to take up his mantel though.
I don't think these mistakes are crazy, though they are mistakes. I remember being puzzled by a similar kind of problem myself. Jeff Speaks brought up the following puzzle: Socrates is not an abstract object, he's concrete. And the set singleton Socrates contains Socrates, right? So how can it be an abstract object, since one of its parts is concrete? Similarly, Russellian propositions are supposed to be abstract objects which (may) contain concrete ones. But how could they be? (Didn't Russell and Frege have an argument about this too? I'll get the reference from Bernie in the morning and see if Frege was worried about the same thing.) As I recall, Michael Nelson made quick work of us. His response: you're only worring about this because you're thinking of the set membership relation as being like the physical containment relation. But that's not what it is, and there's really no problem with it holding between conrcrete objects (like Socrates) and abstract ones (like singleton Socrates, or the proposition that Socrates is mortal.)
Posted by: Gillian Russell at May 4, 2005 01:12 AM
I see I'm a good 7-8 months out of date. But just in case you're still thinking about this stuff, I always thought the following passage from RFM was interesting (and it suggests that we may not need to generalise in the way Mill wanted to in order to run into problems):
"……This [by learning to count different collections of physical objects] is how our children learn sums; for one makes them put down three beans and
then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, then that would be the end of all sums.
“But shouldn’t we still then still have 2 + 2 = 4?”—This sentence would have become unusable."
(RFM I, §37.)
According to the notes I made when I was reading this stuff, there's a non-mathematical example making a related point at Investigations §80, though I can't remember in what its non-mathematicalness lies.
Go Coffa!
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at December 18, 2005 04:39 PM