### Impossible Beliefs

In the Metaphysics Seminar an issue came up that is confusing me. The issue is what is it it possible to believe and what is it possible to know. In particular is it possible to believe something that is necessarily false? Or what amounts to the same thing, if I believe that p then is p possible? The intuition changes sharply in the third person case. It seems obviously true that S can believe that p where p is impossible. This difference in first person and third person ascriptions makes me suspect that "possible" "impossible" "necessary" are propositional attitudes. Moorean paradox can be seen as the mark of the propositional attitude. It is nonsense to assert "I believe that p but it is impossible that p" "I know that p but it is possible that ~p" Interestingly this works across the knowing how and knowing that distinction. So "I know how to ride a bike but it is impossible to ride a bike" is contradictory, as is "I believe I can ride a bike but it is impossible to ride a bike".

The example which came up in the seminar was that Pythagoras believed there were two non zero integers such that the ratio between them was the square root of 2. During his life time this was demonstrated not to be the case. So therefore Pythagoras believed something that was impossible.

But my contention is that this is just another way of saying that Pythagoras believed something false. Mathematical truths are timeless. So it will never be and was never true that the square root of 2 is the ratio between two integers, and we know that now. But I guess it is a fairly standard view that all propositions are timeless in this way. So it will never be and was never true that I had kippers for breakfast this morning. And from my perspective now, it is impossible that I did. But yesterday evening it was possible. My point is that just because something is timelessly true doesn't give it the special status of necessarily.

But maybe mathematically proven truths are necessary, not because they are timeless, but because they are mathematical. What is mathematically proven is true in all mathematically possible worlds. But what about mathematical theorems that have yet neither been proven nor disproven? Are they possible in mathematically possible worlds? There is no way of telling. If there is no way of telling, then introducing the notion of mathematically possible worlds doesn't help in elucidating the meaning of possibility. In a possible world I prove that Goldbach's conjecture is false. Is this a mathematically possible world? It seems like we have to find out whether Goldbach's conjecture is false before we can answer this question.

Some say it is impossible to find a needle in a haystack, but this is false. Given enough time and a good methodology, it is possible to find a needle in a haystack. So Pythagoras sets about searching for a needle in a haystack. He believes that he will find the needle in the haystack. It is possible that his belief is true. After Pythagoras has been searching for many years someone invents a metal detector and demonstrates that there never was a needle in the haystack. So in a sense it was impossible that Pythagoras would find one. But is this to say anything mataphysically different from simply that Pythagorus's belief that he would find a needle was false? Or his belief that he could find a needle was, in this case, false? I can't bring myself to believe that his belief was necessarily false in any non epistemic sense. Of course it is impossible to find something that isn't there. Once you realise it isn't there you stop looking.

One good rule seems to be that if something is true then it is impossible that it is false. But another equally valid rule seems to be that it is possible that something is true, were it not for the inconvenient fact that it is false. So we are tempted to think of a more absolute impossibility where something is necessarily false when, even if it were true, it would still be false. The only thing I can think of that fits into this category are self denials like "This sentence is false". It is impossible that this is true, because even if it is true it is false. But unfortunately this just means that it is necessarily true as well as necessarily false. I contend that the only propositions that are metaphysically necessarily false are also necessarily true. I am generalising from one case and my own lack of imagination. So counterexamples please: Necessarily false timeless propositions.

The example which came up in the seminar was that Pythagoras believed there were two non zero integers such that the ratio between them was the square root of 2. During his life time this was demonstrated not to be the case. So therefore Pythagoras believed something that was impossible.

But my contention is that this is just another way of saying that Pythagoras believed something false. Mathematical truths are timeless. So it will never be and was never true that the square root of 2 is the ratio between two integers, and we know that now. But I guess it is a fairly standard view that all propositions are timeless in this way. So it will never be and was never true that I had kippers for breakfast this morning. And from my perspective now, it is impossible that I did. But yesterday evening it was possible. My point is that just because something is timelessly true doesn't give it the special status of necessarily.

But maybe mathematically proven truths are necessary, not because they are timeless, but because they are mathematical. What is mathematically proven is true in all mathematically possible worlds. But what about mathematical theorems that have yet neither been proven nor disproven? Are they possible in mathematically possible worlds? There is no way of telling. If there is no way of telling, then introducing the notion of mathematically possible worlds doesn't help in elucidating the meaning of possibility. In a possible world I prove that Goldbach's conjecture is false. Is this a mathematically possible world? It seems like we have to find out whether Goldbach's conjecture is false before we can answer this question.

Some say it is impossible to find a needle in a haystack, but this is false. Given enough time and a good methodology, it is possible to find a needle in a haystack. So Pythagoras sets about searching for a needle in a haystack. He believes that he will find the needle in the haystack. It is possible that his belief is true. After Pythagoras has been searching for many years someone invents a metal detector and demonstrates that there never was a needle in the haystack. So in a sense it was impossible that Pythagoras would find one. But is this to say anything mataphysically different from simply that Pythagorus's belief that he would find a needle was false? Or his belief that he could find a needle was, in this case, false? I can't bring myself to believe that his belief was necessarily false in any non epistemic sense. Of course it is impossible to find something that isn't there. Once you realise it isn't there you stop looking.

One good rule seems to be that if something is true then it is impossible that it is false. But another equally valid rule seems to be that it is possible that something is true, were it not for the inconvenient fact that it is false. So we are tempted to think of a more absolute impossibility where something is necessarily false when, even if it were true, it would still be false. The only thing I can think of that fits into this category are self denials like "This sentence is false". It is impossible that this is true, because even if it is true it is false. But unfortunately this just means that it is necessarily true as well as necessarily false. I contend that the only propositions that are metaphysically necessarily false are also necessarily true. I am generalising from one case and my own lack of imagination. So counterexamples please: Necessarily false timeless propositions.

## 5 Comments:

I think at least one of us is confused. Some comments below may help us.

I presume you are talking about some kind of logical/metaphysical modalities throughout. In which case it seems straightforwardly true that we can believe things that are necessarily false. For example, Pythagoras’s mathematical belief that you mention and the belief that Hesperus is not Phosphorus that the ancients held. Both of these are false and necessarily so. My intuitions are constant across the first person and third person cases, although I would not at this moment ascribe any impossible beliefs to myself but neither would ascribe any false beliefs to myself.

Moore’s paradox may be related to propositional attitudes but it is too quick to move from the infelicity of “I believe P but ~F(P)” to the claim that F is a propositional attitude. If knowledge is a norm of assertion and you cannot know something that is impossible then this alone explains the paradox.

Ascribing a necessarily false belief is another way of ascribing a false belief but is a specific way of doing so. Impossible propositions are but a subset of the false propositions.

If something is true it is not possible that it is (actually) false but if it is contingently true it could have been false. Given the lack of a needle at all times during Pythagoras’ search he could never have actually found one in that haystack. But it is possible that there could have been a needle in that haystack and that he would have found it.

Some truths, such as the truths of mathematics could not have been false. The fact that reflection on possible worlds does not help us to determine as yet undecided mathematical theses is neither here nor there. Goldbach’s conjecture if true is necessarily true and if false necessarily false.

“Hesperus is not identical to Phosphorus at t” is necessarily false (false in all worlds where Hesperus exists at t and all worlds if we employ a negative free logic) and timeless for any substitution for t at which Hesperus exists.

Thanks for taking the time to address my problems, and it is me that is confused. I accept that mathematical truths are at least in some sense necessary. Identity statements are also in some sense necessary as is the validity of logical arguments. I confuse the necessity of "If p then p," with the claim that p therefore necessarily p. When I make this move I am accused of the modal fallacy. But if I accept that I am making a mistake here, then I am confused when people seem to make this same move in mathematics, or logic and about identity statements. To use an example, let G be the statement that every even number is the sum of two primes. I actually believe this. This means I think that it is true. So I believe it is necessarily true since if G then necessarily G. I am not a mathematician, but under standards of justification that fit my purposes I have very good reason to believe G. No counterexamples have been discovered to my knowledge and I understand that incredibly huge even numbers have been tested. This indicates that G is no coincidence. Even if G doesn't apply to an even number, it is a number so high that it has no meaning in my life. If I wanted to count the atoms in the universe or the permutations of the human genome, I will be safe in the knowledge that G will hold true for these numbers. So by normal standards of knowledge attribution I know that G. But now it seems to commit me to say that I know that G is a necessary truth. But I don't know this. I accept the possibility that some clever mathematician will prove that G is false. So if G then necessarily G doesn't seem to hold.

You could easily say that I don't know G in spite of my arguments. But my justification for G is a lot stronger than my justification for many things that I do know. I know that all UK branches of Tesco's take switch. How is it that I know this but don't know G, that every even number is the sum of two primes?

You might want to say that mathematical knowledge requires proof, not conclusive evidence of an inductive kind. Then we can have the neat arrangement that everything known mathematically is proven mathematically, and anything proven mathematically is necessarily true. I accept this, and down grade my knowledge claim that G to a justified true belief. But it still seems sensible for me to assert that G without asserting that G is necessary. Suppose there is no proof that G. I feel that in this case I could assert that G and assert that G is not necessary and that both assertions could be true.

My question is not about the necessity of maths but about what function modal concepts serve.

In the absence of proof we want to say it could be that G and it could be that not G. Now suppose I assert G, that all even numbers are the sum of two primes to a mathematician, it seems perfectly in order for him to say "not necessarily" and by this mean there is no proof that G. So "necessarily" here has a mathematical meaning. Now suppose I assert P (that the square root of 2 is the ratio of two postive integers). The same mathematician might now say "That is false", if I respond that it could have been true, he replies that, no, it is necessarily false. By this he means that it has been proven to be false. So "necessarily" here means proven. This accounts for the fact that Pythagoras believed something that was necessarily false. It is proven to be false. This intuition is that once something in mathematics has been proven, it has been proven for all time. I accept this. To know that G, it is necessary to discover a proof that G, and since G is necessary if there is a proof that G, then it seems that we are talking of epistemic necessity. G is necessarily true if and only if G is known. "Known" here is timeless.

Now people will object that I am confusing provable with proved. There might be a proof for G that no one ever discovers. In this case G is necessarily true in a non epistemic sense. But I can accept this, it is just that provable is equivalent to knowable.

You say you accept that mathematical statements are in some sense necessary. I think we should stick with epistemic and logical/metaphysical necessities and forget the rest. The orthodox claim is that mathematical truths are metaphysically necessary.

People make the move from P to Necessaily P in mathematics and identity statements because the truth-values of these statements are necessary (according to orthodoxy). Once you have established P you can conclude Necessarily P for these type of statements. That you can do this here does not license the move in other domains.

If you know that if G then Necessarily G then why are you reluctant to say you know Necessarily G when you say you know G? You deny yourself knowledge of this modal fact because you accept the possibility that someone will prove G false. If someone could prove G false (that is metaphysically could and where prove is factive) then you would be correct to say that you do not know Necessarily G, since G would be false in some world and hence it would be false, and unknowable, that Necessaily G. But neither would you know that G since for mathematics if possibly ~G then ~G. So if this is metaphysically possible then this robs you both of your modal and non-modal mathematical knowledge, assuming as we should that knowledge is factive. Whether your willingness to accept this as metaphysically possible, which is not factive, robs you of knowledge in one or both cases is another matter, but your psychological attitude does not threaten if P then Necessarily P for these claims.

However, I suspect that what you take yourself to accept is some epistemic possibility that someone will prove ~G. If this means that for all you know then someone might prove ~G then you don’t know G because if you did know G then this knowledge would exclude ~G and someone’s proving it. There is more to say here but again does not threaten if P then Necessarily P.

You say it seems sensible for you to assert G and not to assert necessarily G. Well maybe. But this would be because you doubt the principle in question if P then Necessarily P.

You say “Suppose there is no proof that G. I feel that in this case I could assert that G and assert that G is not necessary and that both assertions could be true.” Well again this must be because you doubt that mathematical truths are necessarily true. You can hold this position if you like, but I would not base it on the conceivability argument that seems to be driving you to this – the fact that you can conceive of someone proving ~G. I’ll wager that what you conceive here is little more than a mathematician announcing a proof and it surviving some scrutiny. But that is not enough to think that G could be false in some metaphysically possible world.

Then you seem to confuse epistemic and metaphysical modalities. You write “In the absence of proof we want to say it could be that G and it could be that not G. Now suppose I assert G, that all even numbers are the sum of two primes to a mathematician, it seems perfectly in order for him to say "not necessarily" and by this mean there is no proof that G. So "necessarily" here has a mathematical meaning”.

The “could” you appeal to in the absence of proof is an epistemic could. Similarly, if I discovered the person I thought was my father is not in fact my father then David Lewis could be my father. But this is epistemic could. Whoever is my father is necessarily my father. It is not metaphysically possible that a could have been my father as could b where ~(a=b).

I don’t think the mathematician’s necessarily has a mathematical meaning. Here I think it is best interpreted as epistemic. Sure G is not subject to counterexample but that leaves it open whether it is in fact true or not. He is not agreeing with your assertion that G and then adding the caveat that although G not Metaphysically/Mathematically Necessarily G. If you say Man U will win the premiership and I reply not necessarily, I am neither agreeing with your original assertion nor making a claim about what is metaphysically posiible. Rather, I am saying I wouldn’t be so sure.

Of course he is your mathematician so you can have him say what you like, but if he means metaphysical necessity then he like you rejects the move from G to necessarily G.

Thanks Lee, I guess I've drifted into mathematical necessity because I want to deny that there is a tripartate probability: epistemic/subjective or objective. Epistemic and subjective are just normative and descriptive versions of the same thing which can be measured by betting behaviour and rational choice. But people are strongly attached to views that there are objective chances and mathematical probabilities that somehow transcend justification.

I want to say that to have any kind of belief in something, you must assign some probability to it. If you believe that p is necessarily true, you cannot assign any probability other than 1. Any rational probability assignment requires background assumptions that are certain "there is no probability without certainty". If you delete or add assumptions then your probability assignment will change. So prob(H=P) is 1 if you assume that H=P. But if you don't make this assumption then prob(H=P) need not be 1. When people talk about identity relationships being necessary then all that is meant is that given (A=A) then prob (A=A)=1.

So what I am confused about is statements like given G then necessarily G, but given not G then necessarily not G. This is fine, but if I believe that G is more likely than not G, then what kind of probability am I talking about? Epistemic you say. Well, there is more than 1 way of coming to know something. Suppose Goldbach's conjecture was tested for every even number up to the highest number that can have any application outside pure mathematics. I am committed to saying that with this kind of evidence I should assign prob(G)=1. In this case I am warranted in claiming to know that G. But I still may accept that there is a possibility that someone may prove that G is false. I cannot assign a probability to this possibility because my sample size is huge and I can't reason with infinity. The probability that G is false comes out as infinity/infinity which could be any figure. So such considerations don't directly commit me to revising my assignment prob(G)=1. If I revise it it is 1-infinity/infinity. But it does not commit me to to assign prob(there exists a proof that G in mathematical space)=1 either. Without a proof it is not clear to me why G is a necessary truth given the axioms of mathematics.

I believe it's Roy Sorensen who came up with the proof that it's possible to believe the impossible. He think's it's possible, but for a while he was in the minority. He wins.

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