The Empirical Justification of Mathematics

Things have been quite around here lately, mostly because I have been lost in GTA IV and finishing my dissertation. Well I am making some progress (on both 🙂 ) and so will try to get to some of the comments around here.

In answering the comments on the Refutatioin of Rationalism I started thinking about Quine’s indispensibility argument for the empirical justification of mathematics. The argument starts from Quine’s claim that we are ontologically committed to the things which we quantify over in our best theories. The indispensibility of mathematics to physics means that we are committed to the existence of numbers (but not, obviously, to their non-physical existence). So Quine went on to argue that, since our theories all get confirmed or disconfirmed together as a group, the empirical confirmation of physics is empirical confirmation for mathematics. In this way mathematics is empirically justified.

One problem with this argument is that it depends on confirmation holism. That is, it depends on the claim that all of our theories are confirmed or discomfirmed together. None ‘face the tribunal of experience alone’. I then started thinking about how Rosenthal’s version of this argument avoids this commitment and so is a better argument. Sadly Rosenthal has never published this argument (I heard it in a seminar on Quiene and Sellers he gave) so I will try to recreate it as best as I remember.

The basic idea is: if we ever had empirical evidence that some truth of arithmatic was false we would have to admit that it was false. But if so then mathematics is empirically justified. To make the case he asks us to entertain the following scenerio. Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable.

If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.  

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21 thoughts on “The Empirical Justification of Mathematics

  1. Okay, so now you take 6 liters of one liquid and 7 liters of another, and pour them together and count the total liters, and there is not 13 as you would suspect from the sheep example, but instead only 10 liters! What happened? Math is not empirically justified?

  2. Whoops, that should be 7 + 6 = 13; I don’t know whether the other equation popped up from. Some rationalist must have thrown a mathematical object at my computer….

  3. To make things fair to the rationalist, and remove any possibility of new sheep appearing and disappearing, this is how it should be done…

    Count the sheep from the the first pen, then when you start counting the sheep from the second pen, also in parallel CONTINUE counting the sheep from both pens…

    So, we count like this… 1, 2, 3, 4, 5, 6 and then continue by BOTH counting the sheep in second pen but also continue counting all the sheep, so we get 1,7; 2,8; 3,9; 4, 10; 5, 11; 6, 12; and finally… 7, 13

    If one knows to count, there is no way this can turn out different!

  4. Oh no, I’ve been counting the wrong way for years!

    But, Tanasije, if you count how many the parallel number there are, you get 14!
    1, 7 (2), 2, 8 (4), 3, 9 (6), 4, 10 (8), 5, 11 (10), 6, 12 (12), 7, 13 (14)

  5. > If one knows to count

    Tanasije,
    So in our ‘Richard Brown’ world you have just provided us with the conclusive proof that no one knows how to count 🙂

    My theory is that the extra number always gets added just before you try to start counting the second pen.

  6. You describe a case where a mathematical claim is called into question by experience. It is not clear that this case is metaphysically possible, so I think it risks a stand-off. I would suggest the more plausible case, employed by Kitcher, where an agent decides to reject a mathematical theorem after having been told by an expert mathematician that the proof contains a flaw.

    Still, both sorts of cases assume that a priori justification cannot be defeated by experiential factors. This is too stringent a requirement, I would suggest. For arguments that genuine a priori justification can be defeated by experience see Casullo’s book.

  7. “If one knows to count, there is no way this can turn out different!”

    Why not?

    Everyone keeps saying that, but no one has given an actual argument for this claim.

    The situation I described above is clearly conceivable and so should be metaphysically possible. But if not what we need is an argument showing either that it isn’t conceivable or is but isn’t really possible. No such argument has been given.

    Brandon, the answer is ‘Neurath’s Boat’

  8. Brandon, the answer is ‘Neurath’s Boat’

    Which is not an answer at all; it’s as if you asked how the rationalist thought we could establish necessary truths, and they answered “Plato’s Cave”. Moreover, Neurath’s boat is not infinitely elastic; if you’re at sea you can repair a leaking boat, but you can’t repair a boat that has already sunk. Part of the boat is the scientific method itself, which in order to be what it is has to cohere with a great many things already in place; the question here is precisely what it means to have a scientific method that is established entirely independently of even basic mathematical principles but at the same time is sufficiently sophisticated that it could unambiguously provide a filter of what mathematical principles most cohere with the boat.

    Look at it another way: what makes your scientific proof that 7 + 6 = 14 a proof that 7 + 6 = 14 rather than a reductio ad absurdum of scientific proof? Only if scientific method is foundational and mathematical principles are not; but in that case ‘Neurath’s Boat’ can’t be the answer: Neurath’s boat has no foundations, it’s just there, and being fixed piecemeal. What you are suggesting would require ripping out vast segments of the boat, and the coherentism of the boat image has no room for such radical revision. Some ground or platform for radical revision has to be admitted; and what you are suggesting is that scientific method can provide such a platform; in which case the question still stands: what does it mean to prove something scientifically in a context in which one does not already know even the most basic mathematics?

  9. Brandon,

    Interesting comment. Perhaps we are using ‘scientific method’ in slightly different ways? I mean by it simply the view that truth is discovered via repeatable exerience. This is established entirely independently of basic mathematical princilples and can provide a filter for what basic mathematical pronciples most cohere with the rest of the boat. I don’t really see what the problem is supposed to be here. We have done the same thing for logic. The history of Quantum Mechanics is in part the history of teh question of whether or not we should revise our standard logic so as to not include the law of the excluded middle. This debate still goes on (having just hired Graham Proest at the Graduate Center, I figure I have to mention that Dialethism is alive and well). You may ask ;how can they debate this when logic itself is in question?’ They manage. I would argue that logic is more basic than mathematic (I am still convinced that we can reduce mathematics to logic) so it seems to me that if the debate makes sense about logic, then there is no problem when we turn to mathematics.

    “Look at it another way: what makes your scientific proof that 7 + 6 = 14 a proof that 7 + 6 = 14 rather than a reductio ad absurdum of scientific proof? “

    That’s a good question and I see where you are going with it. This is your point that most (if not all) of the arguments here are radically question begging. So, you go on to answer your question by saying,

    “Only if scientific method is foundational and mathematical principles are not”

    But as you correctly point out, the Neurath’s boat metaphor suggests that there is no foundation. This is exactly the point of the Quinian kind of holism that the Neurath boat is supposed to invoke. But the claim isn’t that some one ofthese is foundational. Rather we must ask ourselves which of these we ought to keep. That is, given this kind of repeated experience we cannot both (if we keep the law of noncontradiction, that is). As you say, revising mathematics would requie revising a great chunk of the boat. No doubt about it. And we surely need a platform, and as I have suggested the platform is the scientific method and basic logical principles (ignoring for now the question of their revision…I personally don’t think there is any reason to revise them that we have yet discovered).

    In short there is nothing wrong with claiming that the scientific method and the basic logical principles (which are both empirically justified theories) serve as the basis for our empirical proof that 7+6=14.

  10. Richard,

    When I said ‘if one knows how to count’, I was thinking of the counting example I gave – that is, where one counting goes this way – 1, 2, 3, 4, 5, 6, and then count (1, 7), (2, 8), (3, 9), (4, 10), (5, 11), (6, 12), (7, 13). It is the way to be sure that we are counting *one and the same thing*.

    So, when I say, if one knows how to count, the result can’t be different, I’m saying that if we agree that this is the sequence – 1, 2, 3, 4, 5, 6, and then count (1, 7), (2, 8), (3, 9), (4, 10), (5, 11), (6, 12), (7, 13) which will be applied, there is nothing that can be different! If you want to say that 6+7 might be 14, then provide example of how it could happen in the way of counting that I proposed. But, if you change that sequence, it will be not because of something that happened in the world, it would be merely negating the established sequence of numbers, and that is what I would call “not knowing how to count” 🙂

    You may ask should we agree to my way of counting, and not yours. But that doesn’t help your position, because I can state that *that is* what I think is the meaning of 6+7=13, and as far you can agree that that *could* be its meaning, you will have to agree that there are *possible* statements, whose truth can’t be different.

    As a further argument, that my way of counting is the right one if we are to see the mathematical truths in the real world phenomena, I can say that if we don’t have this kind of criteria – i.e. that it has to be one and the same set which we are counting, we might as well count the number of sheep in the first pen as 6, then count the number of sheep again, get 7, and proclaim that 6=7.

    Would you go so far, in saying that if this happens consistently, we would have empirical proof that 6=7?

  11. Just further thought. Imagine that we count the number of sheep and get 6, then recount them and get 7. Now, imagine that we recount the first 6, and then recount them, and get 7 again.
    We will now have a procedure by which we can get as much sheep as we want, just by counting the first 6 🙂

  12. If we are talking about simply repeatable experience, then the question still arises: on the basis of what are we to take the repeated experience of adding 6 & 7 to get 14as evidence of the truth of 6 + 7 =14, rather than as an evidence that our experience is wonky? (A line of reasoning: If 6 + 7 = 14, it follows that 6 = 7, and if 6 = 7 it follows that 0=1. Since 0 + 0 + 0 = 1 + 1 + 1, it follows that zero experiences is the same number of experiences as three experiences, or, indeed, any number of experiences. Therefore all truths can be established a priori, on the basis of 0 experiences. But if anything can be proven a priori, it’s that 0 is not equal to 1. Moreover, the same problem still arises with repeated experience as for more sophisticated scientific reasoning. How do you prove anything on the basis of repeated experience if you can’t even tell that 0 experiences is not the same as 1 experience?)

    A similar problem arises across the board; suppose the law of noncontradiction can only be an empirically justified theory. How do we empirically justify something if we can’t even assume that an empirical justification of the law of noncontradiction is not also an empirical justification of its falsehood? And so forth for all major principles of logic and mathematics.

    Dialetheism et al. is not really to the point, because even dialetheism is actually a conservative modification of classical logic; it just allows discussion of domains in which contradictions are posited. But making sense of dialetheism presupposes being able to make sense of domains in which contradictions are disallowed. Thus it’s an entirely different debate.

    My point about Neurath’s boat was that it only admits of a relatively conservative coherentism; massive changes have to be justified some other way than simply coherence, and so simply appealing to Neurath’s boat doesn’t handle the problem the rationalist is going to propose.

  13. Sorry for the delay in getting back to your interesting comments Brandon!

    This is hard to talk about because we haven’t had the kind of experiences that we are here talking about. So, we don’t know, for instance, if this is something that would be limited to basic arithmatic for numbers totaling more than 10 (and say, the math truths stayed the same for 1+1=2, 1+2=3, 2+2=4, etc) or if this was indicative of some more wide-ranging issue in arithmatic. You seem to suppose that if ^+7 doesn’t equal 13 then 1+1 won’t equal 2 but that itself, according to me, is an empirical issue. So whether your line of reasoning goes through or not is itself an empirical matter.

    Consider an analogy. Entaglement in quantuum physics was first introduced in an attempt to illustrate, a priori, the absurdity inherent in quantum theory. If true the theory predicts that interacting with one individual object instantaneously effects the entagled partner no matter how spacially segregated the two particles may be. This seem to call into question our ability to dinstinguish one particle from two particles. But the evidence since then seems to suggest that this is actually the way that things are (this is sort of ironic since Einstein’s theory had done the very same thing for non-Euclidean geometry). But has this discovery led to any of the results that the rationalist predicts? No, There is, oc course, a question of how this new discovery has to be accounted for. Do we still count the entagled particles as distinct particles and so say that there are two objects there? Or do we say that there is really only one partile there, even though it is comnposed of parts that are spatially segregated? We don’t, as yet, have an answer to this debate that I am aware of, but it certainly hasn’t rendered it impossible to individuate experiences or to question the scientific method.

    So, we don’t know how extensive the revision would have to be in the imagined case we are considering, but we can see from some actual cases that major revisions can be considered so it is in principle possible to do the imagined kind of revisions.

    Saying ‘but we always need to make sense of some domain where things are ‘normal” is just to say that we can’t revise the entire boat all at once…but a complete over-haul could happen substantial-chunk-by-substantial-chunk (like, I imagine, the shift from Aristotealianism to the modern mechanistic view to our micro-physics…

  14. You seem to suppose that if ^+7 doesn’t equal 13 then 1+1 won’t equal 2 but that itself, according to me, is an empirical issue. So whether your line of reasoning goes through or not is itself an empirical matter.

    I don’t think there needs to be any supposition; the real question is how we can have a consistent account of building up any empirical issue out of repetitions of experiences if we cannot even presuppose basic number relations as a logical precondition for repetition or, indeed, experience. I’m not really convinced that your quantum entanglement is a good example; it’s really just a more complicated version of a question like “Is what we call Hesperus the same thing as what we call Phosphorus?” But how we would even understand such questions about sameness if we could not presuppose some notion of ‘one’ is a mystery, and how we can identify anything as a repetition without already being committed to basic arithmetic is equally so. Thus the empiricist account itself seems to assume basic principles of arithmetic as unrevisable in order to make coherent sense of this ‘experience’ that they keep talking about.

  15. Sure, that is the way it seems to you Brandon because we haven’t had the kind of evidence we are talking about and so haven’t had to really deal with how we would deal with it.

    But really Brandon I get the feeling that we are passing in the night here. I agree that basic arithematic is assumed by the empiricist’s account of how we know. But this is justified because of a long history of certain regularities in nature/our experience of nature. To find evidence now, after so long a regularity, does strike us as unlikely. It strikes me as unlikely as well. But the question here is whether this obvious truth is necessary and known independantly of any experience and so in principle cannot be disconfirmed by experience or whether it isn’t necessary and is known only through experience. The argument here is simply that if we had this kind of experience we would have to admit that a certain mathematical truth was false and so the justification of mathematics is ultimately our experience. You object that we would not be able to make sense of this kind of revision since we could no longer tell when we had one experience or two or when something was repeated. But this is an assumption, as I tried to point out in the last post, that is unsupported. In the scenerio I envision we would certainly have to revise our system of mathematics, but again we do not know how extensive this revision will have to be. Either way we will still be able to make sense of there being ‘one experience’ (though what counts as one may change). So too we will still be able to make sense of addition, though not in the form that you now know it. Addition as we know it is wholly and totally captured by the Peano axioms. But these are just one set of axioms from an unlimited number of possible axioms for other functions (schmaddition, etc). If Peano’s axioms did not formalize a system that matched or caprtued the empirical truth no one would care about what he calls addition. The Peano axioms offer one way to define ‘addition’ in terms of ‘successor’ etc but that just means that arithmetical truths are all anayltic, not that they are knowable a priori.

    So to summate; what the empiricist has to assume is that we will still be able to make sense of mathematics in light if this radical discovery…you haven’t yet given any reason to think that this won’t be true.

  16. what the empiricist has to assume is that we will still be able to make sense of mathematics in light if this radical discovery

    I’m not sure why you think this is true; if the empiricist has to assume that we will still be able to make sense of mathematics, this sounds very much like the empiricist has to treat certain (very general) mathematical principles as unrevisable — if not, say, basic arithmetic, perhaps something more foundational, like certain principles of category theory. But both you and the rationalist seem to agree that the empiricist can’t take anything to be unrevisable, because the unrevisable is in rationalist territory.

    I’m also a little puzzled about your complacency about analytic truths. Although they tend to prefer not to do so, empiricists can have a priori knowledge — they would just interpret it as instinctual, or hardwired, and would simply have to have a way of explaining how and why it’s hardwired. It’s the analytic that has always seemed to give them trouble.

  17. “if the empiricist has to assume that we will still be able to make sense of mathematics, this sounds very much like the empiricist has to treat certain (very general) mathematical principles as unrevisable”

    It may sound like it, but it isn’t. Somethings will have to be ‘held constant’ while the revision is taking place; that much is true. But this doesn’t mean that the things which were held constant MUST be held constant at all time. Once the initial revision is done it may turn out that we need to hold the newly revised stuff constant while we revise the previously held constants. So each thing is revisable, but not all at once.

    “Although they tend to prefer not to do so, empiricists can have a priori knowledge — they would just interpret it as instinctual, or hardwired”

    I would not call that hardwired knowledge a priori knowledge IF it is hardwired in response to ancestreal empirical experience. Real a priori knowledge is immediate apprehension of a necessary fact about reality. What you are suggesting is an explanation of how it could seem to us that we had this kind of knowledge when we in fact do not. That is not having a priori knowledge!

    The reason the analytic has given the empiricist trouble is because of the link between the a priori and teh analytic, but that can be severed. As I have elsewhere argued, I think that analytic statements are themselves revisable. So I have a ‘weakened’ sense of analytic in mind.

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