If someone demanded proof that 1+1=2 is true what would you do? Would you get an object and place it next to another object and then count them? One might argue that that doesn’t prove that 1+1=2 because 1+1=2 is a necessary truth and you cannot get necessity from experience (as per the history of philosophy). If one thought that all knowledge comes from experience one might then, like Quine, think that it is possible that we could have experience that dis-confirmed mathematics. For instance David Rosenthal has argued that if we ever had irrefutable counting evidence (i.e. widespread, re-created and independently confirmed) that 1+1=2 were false then we would have to admit that it was false and so mathematic is contingent on experience. Yikes!

Or would you appeal to the Peano axioms, which include the claim that 0 is a number and that it has a successor denoted by S(0) and then define addition as

(for all a) a+0=a

(for all a and b) a+S(b)=S(a+b)

So then it is easy to show that 1+1=2 as follows

1+1=a+S(0)=S(0+1)=S(1)=2

But then one might worry about the successor relation. It might seem as though we have simply assumed addition in the definition of the successor relation (i.e. it tacitly assumes that S(a)=(a+1)).

Of course we can show that S(a)=(a+1) by the definition of addition above. So, let S(0)=1 then a+1=a+S(0) by the definition of addition we get a+S(0)=S(a+0) and since a+0=a we get S(a) so (a+1)=S(a). But then we seem to have assumed addition by stipulating that S(0)=1 (as we actually did in the initial proof of 1+1=2…worse and worse!)

Is the claim that when you have nothing and add a thing you then get only the one thing (i.e. S(0)=1) supposed to be a truth that we just apprehend with pure reason? A self-evident truth that is ‘clear and distinct’? Or something that experience has trained us to believe? Is there any non-question begging reason to prefer either of these?

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Hi Richard,

I think the best way to capture the intuitive necessity of 1+1=2 is to say that…

Whateveris 2 is also 1+1, andwhateveris 1+1 is also 2. So, to avoid mapping “+” to some empirical acting – instead the whole 1+1 is to be taken as a concept.I think of that this way. If there is a pair of things in front of us (or pair of things imagined, or assumed, etc…) we can either a)put attention on the pair qua pair or b)put attention on each of individuals as another to each-other. I think that a) corresponds to something being determined as 2, and b) corresponds to something being determined as 1+1. While there is the pair in front of me, I can switch my determination of it between a) and b) in similar way as it happens with the Necker cube. Somewhere there I think the necessity is “seen” (or intuited/comprehended).

So, I think that what happens in 1+1=2 is not identifying the left and right side of it as something separate, but as necessity if something determined as 2, it will also be possible to determine it as 1+1 (and the other way around).

BTW, I don’t think this is truth based on the definitions on the terms, or some formalism. I think a person can know what ‘1’ is and what ‘2’ is, and not know if 1+1=2. (Of course that doesn’t mean that those symbols can not be used in a formalism, that would map in some way to the intuitive understanding, which I think is the case with the idea of axiomatization of math)

“So, to avoid mapping “+” to some empirical acting – instead the whole 1+1 is to be taken as a concept.”

That’s exactly what I’d say. 1+1 doesn’t “give rise” to 2ness, rather, we define it as meaning the same thing as 2, thereby avoiding any “prove it” objections!

Furthermore, “1+1” is contained within the definition of “2”. We interpret “1+1=2” as one single concept, not anything that requires an empirical bridging of sorts. 🙂

Hey Tanasije,

So, let me ask you this. Suppose that we were not able to put attention to a pair of things or to them individually, would 1+1 still be 2?

Also, say that we find a pair of objects that we can ‘determine’ as individuals but not as two, or vice versa some objects that we could determines as 2 but not as 1+1 would it still be the case that 1+1=2?

Yes, 1+1=2 (in the sense put forward) would be true even if there were no humans. That the same thing is in same time both a) a multitude/a pair, and b) individuals which are other to each-other, isn’t dependent on our ability to become aware of it qua a) or qua b).

So, for the second question, I would claim that you can’t find pair of objects which wouldn’t be in same time individuals which are other to each-other. There is no way something can be a pair, and in the same time there not to be individuals which are other to each-other.

I think if I were asked why dos 1+1=2, I would start by pointing out that, in both instances of (1) + (1) = 2, here we have 1 being accepted in its definitiveness of value/quantity existence. It therefore meets all rational and logic that when its definitiveness of value/quantity existence is increased by its similarity the definitiveness of value/quantity existence clearly cannot be the same. This definitiveness no longer enjoys a singular value/quantity existence. This then forces us to accept a new definitiveness of value/quantity existence , i.e. 2

This principle of value/quantity definitiveness applies to all numbers. I say numbers, because I don’t accept zero [0] as a number since it has no definitiveness value/quantity existence.

Ok, so my question is why would it still be true? What is the truth of (a) and (b) dependent on?

As for the seoncd claim, again the question is why not? Why couldn’t we find such a pair? Is it because we couldn’t have that kind of experience? Why couldn’t we? For instance, doesn’t non-locality suggest that something can be a pair and yet not be ‘other to each-other’? So if the answer to the first question is ‘something about experience’ then you have to admit that you might have experience which contradicted 1+1=2…on the other hand if you say ‘something about reason’ then the problem is that there is no non-circular way to establish (a) and (b)

The truth of (a) and (b) separately depends on the state of affairs.

e.g. ‘X is a pair’ is true iff X is a pair. (Of course this abstracts from what kind of pair it is, e.g. pair of frogs, pair of books. The pair is always pair of certain things which fall under common concept (frog, book, etc..), so properly speaking the multitude is pair of frogs, pair of books, and never pair simpliciter, but for simplicity sake let’s keep talking about ‘pairs’).

“whenever something is (a) is also (b) and other way around” is though true because of the meanings of (a) and (b), and not contingent facts in the world (though it can be applied to the contingent states because of ‘whenever something is…’). I think we can comprehend that necessity (in the way described), and that there is no need (nor sense) to ask for further ground (e.g. formalisms or Kantian categories).

The idea is that the goal of philosophy is to comprehend necessary relations between notions, and that anything ‘further’ would be just asking for trouble (i.e. obscuring what has already been understood).

So I understand that whenever there is a pair, there is a thing and another thing, and I deny any need to go further in the analysis of ‘where this truth comes from’. I guess that this kind of “1+1=2 is true damn it!” argument isn’t worth much, but I hope it makes sense with the previous point I made about how when we understand something we need not ask further, and really that there is no sense in asking further. Of course there is always possibility for mistake. We might just think that we understood. So, I might be wrong that 1+1=2 🙂

We can’t find pair of objects which are in the same time not individual and one more individual, because ‘whenever something is a pair, it is also individuals other to each-other’. I understand it that it is so, and think that there is no sense in looking for further ground of that understanding. (Not very proud of understanding that 1+1=2 BTW 🙂 )

I think that is great point with non-locality example. I also don’t think that in the case of entanglement the particles carry their identity through the entanglement. But because of that I also think that one can’t talk about a pair there. There were two individual particles before the entanglement and after the collapse, but as long those are not present as individuals through the entanglement why talk about pair at all? Pair of what? They are not individuals. Why not speak then of a system described but certain wave-equation, which might be collapsed to a pair of particles. But in the same moment they become a pair, they do so by becoming other to each-other (i.e. individuals).

How is 1 + 1 = 2 proved? Maybe that depends upon the context in which a proof is asked for. As far as I am concerned, 2 is equal to 1 + 1 by definition, and so the very idea of a proof of that strikes me as prima facie silly.

But presumably someone else might have learned the meaning of 2 differently. E.g. they might have associated 2 with pairs of similar tokens, such as XX and OO. They might therefore think that 2 is by definition connected with pair-sets. And if their set theory was like NBG they might then think that there could be ones (classes) such as the class of the ordinals, and the class of the cardinals, which did not together make a two (since proper classes cannot be members of other classes in NBG). For them, not only would 1 + 1 = 2

notbe true by definition, it might not even be true at all.But therefore that example (of a possibility) makes me wonder how such people

wouldprove that 1 + 1 = 2? How would they, for example, rule out the possibility that a Cartesian demon had fooled them into thinking that it was true, when really it was not (for all that the true way of such things would not then be for them imaginable)?Maybe, if the context of the request for a proof was a legal or scientific setting (or if one’s philosophy was Brouwerian or Quinean) for example, then a proof might involve showing things like XX and OO, in order to show that 1 + 1 = 2 was very plausible, was highly probable. Although you can’t get necessity from experience, you can get (i) proofs of high probability (or rather, it is often assumed that you can) and (ii) you can sometimes get a clearer sight of a necessary truth by considering well-chosen examples of its instances.

Anyway, I agree with you, Richard, about our assuming an identity in the proof via the Peano axioms, as it seems that it it would be my knowledge of the properties of the natural numbers that would make me believe that they satisfied the Peano axioms in the first place. If I did not already have reason to think that 1 + 1 = 2 then such a proof would not seem to be saying anything about 1 and 2.

Hi Tanasije,

So if I understand you, you think that philosophy is conceptual analysis (“the goal of philosophy is to comprehend necessary relations between notions”) and 1+1=2 is an analytic truth (““whenever something is (a) is also (b) and other way around” is though true because of the meanings of (a) and (b), and not contingent facts in the world”)…but there are some problems with this view…

1. since (a) and (b) are both contingent and learned from expereince (i.e. depend on the state of affairs) it will be a contingent truth that what we count as a pair is a pair or that what we count as one is one, but that seems extremely unlikely!

2. You say that the goal of philosophy is to get at ‘necessary relations between notions’, how do we know that the relations are necessary? You mean the ones that seem true to you based on the experience you have had? Hmmm…that doesn’t seem necessary…There is a legitimate question here. I don’t see how or why asking for more is asking for trouble. I want to know why this thing is true…telling me not to ask and then it won’t bother me that there is no acceptable answer doesn’t help….the idea that ‘when we understand something we need go no further’ seems increadibly destructive to me. It is the attitude that the Scholatic Aristotelian philosophers had in the 16th and 17th Century. That is until Descartes, Newton, Galieo, Boyle and Locke (and so on) mathematised physics. To paraphrase Aristotle, the goal of science, and philosophy, is not merely to show

thatsomething is the case but to showwhy it has to be the case. I just can’t believe that this basic desire to know that all people have is something that we should ignore.3. Finally, the idea that the mathematical truths are analytic is, I take it, what is Logicism (i.e. the idea that the math truths are derivable from the axioms of logic)…This was Frege’s project and though for a while it was thought defeated because of Russell’s discovery of the paradoxs, it has recently made a revival. Crispen Wright is perhaps themost well known defender of a modern kind of logicism. He argues that there is a way to derive the Peano axioms from (second-order) logic in a way that avoids the problems with Frege’s original attempt (involving something that is often called ‘Hume’s principle’, which is I think that the number of A’s is equal to the number of B’s iff there is a one-to-one correspondance between the two groups.).

I sort of think that this is right and so the mathematical truths will turn out to be analytic (except in order to be right numbers have to be abstract oibjects, which pisses me off). But I also think that the Peano axioms don’t show that S(0)=1 or S(1)=2. They show that every natural number has a succsessor that is a natural number but not that it must be ‘the next one’…indeed! How would you define ‘next one’ so that it is exactly one away? The only way to do that is to assume that S(a)=a+1, which is to assume arithmatic in our proof of arithmatic…

Hi Enigman, thanks for the comment.

Some of the stuff I was saying to Tanasije in #3 above I think applies to you as well in so far as you think that 1+1=2 is analytic (i.e. true by definition). Something that occurred to me after I read your comment is that taking the neo-logicist strategy doesn’t help because it assumes that numbers are abstract objects and that we already know what their properties are, so we can know when some number, as you say, satisfies the Axioms, or whether Hume’s principle can be used to define what the numbers are…but this claim is clearly question begging!

So we end up back in our original prediciment…Either 1+1=2 is true because it correctly describes the relationship between the numbers (i.e. the abstract objects that exist necessarily and so on) or it is true because of the formalism that we adopt, but not because it describes anything physical or non. So the issue of whether 1+1=2 is analytic or not and the issue of whether numbers exist or not are independent of each other (The answer can be ‘yes’ in both cases).

Hey Richard,

I’m quite a rationalist, so while I think we learn notions in (a) and (b) based on experience, I don’t think that the comprehension that whatever is (a) will necessarily be (b) is based on experience. So, I’m not saying that those relations are necessary because they seem necessary to us based on experience. I’m saying that we can become aware of the necessity qua necessity. That we can intuit the necessity.

Don’t get me wrong… I’m all for asking ‘why?’ questions. But surely the asking of ‘why?’ questions makes sense only if we allow that on some level there will be some truth that we can comprehend as true, without a need for further justification? I take it that this means that at some point we need to recognize a truth as truth, and to recognize that there is no need for further asking ‘why?’.

So for me 1+1=2 (understood in the sense I described it) is such truth, and I’m quite satisfied with my comprehension of that as truth – as a necessary relation between (a) and (b) – i.e. that whenever something is (a) it will be (b), and the other way arround. A necessity that I can “see”.

Of course taking such stance about this issue, I can’t give further arguments, other than simply claim what I comprehend that 1+1=2, and to ask you to consider (a) and (b) and see if you also acknowledge the necessity of that relation. So, what I think is needed in case of 1+1=2 is not answering ‘why?’, but to comprehend the necessity in right way.

When I mentioned asking for trouble, I had on mind Frege, and also works like Principia Mathematica and set-theory in general, and on other side Kantian project of giving ground to ‘synthetic a priori truths’ (or similar psychologistic explanations). Why(!) would I be satisfied with those complex theories instead of the direct intuitive understanding that 1+1=2? I’m sure those theories open more ‘why?’ questions than the ones that they “solve”.

You say :”To paraphrase Aristotle, the goal of science, and philosophy, is not merely to show that something is the case but to show why it has to be the case.”

If we don’t accept that there are any truths that we can directly intuit, that would become impossible, it would be an neverending sequence of ‘why?’. I would merely put ‘understand’ instead of ‘show’ there, and claim that 1+1=2 is a proposition that we can understand to be the case.

I think your a slightly missing the point. How do you know that what you call ‘seeing the necessity’ might not, for all you know, be the result of a lot of experience…how do you know that it isn’t? Just saying that you are ‘satisfied’ with your understanding (or whatever) it is just avoiding the problem. Whether or not there will be a foundational truth that we can just intuit is what is at issue…So, I say, given the kind of experience that we have had on this planet, and our evolutionary history then OF COURSE it will seem necessary to us but what I want to know is if it really is necessary. What could actually answer that question that isn’t question beging?

Richard,

Technically is not avoiding the problem, but negating that there is one. 🙂

Anyway, back to your original question… You ask what reason could be there to pick intuition over experience (or other way around) which is not question begging.

But, what form would this ‘reason’ take? It should be in form Y:’we know 1+1=2 a priori because X’. But now the empiricist asks “And how do you know Y?” Of course, if one doesn’t accept that something can be genuinely understood/comprehended in it’s necessity, this can’t stop ever. We could take the simplest conclusion based on modus ponens, and such person can ask again why do we think that that conclusion is true.

What would empiricists (of such kind that doesn’t admit intuition) answer to a claim that somebody intuits the truth (e.g. that 1+1=2)? That it is impossible? But isn’t such strong claim for such kind of empiricism? One can ask her for a reason for her claim. If she can’t produce necessary reason then she can’t claim that it is impossible that someone can intuit that 1+1=2. So, if intuitive understanding of truth is possible, why not take someone’s word for it?

Sorry for little disconnected talk, but I find the question very interesting. Thanks for raising it!

What do you mean by “true”?

It’s true that the symbol-string: “1+1=2” is a theorem of arithmetic.

It’s true that arithmetic, as a conceptual framework, makes accurate predictions about some experiences.

Do we need something more?

No worries Tanasije!

You ask what an empiricist would say to someone to somone who claims to have an intuition of an a priori truth that is necessary? I would ask them why they think that the properties that

seemto be necessary, must be…especially when we can imagine that you feel that way mostly as the product of a long evolutionary ‘training session’ in how things work around here. So maybe for creatures like us these things will seem this way even if they aren’t. Earlier I made a compairison to theories of consciousness. So some people hold that when we have a pain we have an experience that we have direct acces to and which we grasp in some special way. When we think about the experience of a pain it seems to us as though all there is to the pain is the painful experience and that seems like it could exist independently of any brain state or whatever. But some naturalistic theories of consciousness argue that what seems like an intrinsic property of teh pain is really a relational property of the pain…or that a conscious pain is a pain that I am conscious of. So I think that what seems like an intrinsic property of the pain (that is, it’s painfulness to have) really isn’t one. It is the result of my being conscious of some first-order pain state. So if you claimed to know that it was necessary that the painfulness of the pain is an intrinsic property of the pain I would say that you are begging a very important question…so why isn’t something like that going on here in this debate?Hi BB,

Thanks for the comment!

I think that is a good question…I take it that what we want to know is whether there is any sense in asking if it could be false…is there a possible world where 1+1=2 is not a theorem of arithmetic? One where it does not make acurate predictions about some experiences? Some people have even thought that the actual world was such a world…are they right? If not how could we possibly show that?

[…] an earlier post (Why Does 1+1=2?) I argued that there is nothing that could possibly decide between whether or not mathematics is an […]

I don’t claim to have any background in the philosophy of mathematics (in my undergraduate term, no course was ever taught on it). But I do have an interest in it.

I am not for sure about this, but it looks as though in asking ‘Why does 1+1=2?,’ you could be asking two different questions. One question would be ‘Why in the formal system of mathematics does 1+1=2?’ Or you could be asking ‘How does a set that has the property oneness combine with another set that has the property oneness to become a set that has the property twoness?’ I think if the question is posed in this way, it seems to me, at least on the face of it, that the problem is not so difficult.

I will take the second possible question first. I am tentatively a naturalist when it comes to the issue of numbers, and it seems to me that numbers are properties of sets. And as properties of sets they are a kind of way of indexing entities in a set. And it depends on what one takes to be a possible set and a possible entity in a set as to whether or not that set has that property. Suppose we see a dog in a yard. If we take this dog to be the only entity in a set (call it ‘dog-set1’), we then say that dog-set1 has the property oneness. I say it’s a sort of indexing because in saying that a set has the property oneness, it is something akin to demonstrating through abstracted ostension that the only entity in it is ‘that,’ whatever entity I’m referring to, in this case a dog.

Now suppose adjacent to our hypothetical yard is another yard where there is a dog. The dog in that yard is of dog-set2, we could call it, which, like dog-set1, has the property oneness. But now upon viewing dog-set2, we now take both dogs to be of one set, say dog-set3. So by conjunction ‘that dog’ and ‘that other dog’ are taken to be part of the new dog set, dog-set3. So the answer to the question, ‘How does a set that has the property oneness combine with another set that has the property oneness to become a set that has the property twoness?,’ is this: it does so by the conjunction of each abstracted ostension. We just take each ostension or way of indexing in this case to be equivalent to what we mean when we say that something is ‘one’ or has the property oneness, and the ostensions or ways of indexing together being equivalent to what we mean when we say that something is ‘two’ or has the property twoness.

This, I realize, is a pretty complicated story to tell, but at least for the moment I think it’s sound. And if the account is true, then we have an easy way to answer the first sense of the question, that is, ‘Why does 1+1=2?’ as ‘Why in the formal system of mathematics does 1+1=2?’ If we take numbers to be a kind of abstracted ostension or abstracted manner of indexing, then we see that this talk of ones and twos is just so much discourse in the formal system of mathematics. Suppose we abstracted our manner of indexing in another way: the ostensive ‘that’ and ostensive conjunction of ‘that’ and ‘that’ to correspond to S and Q, respectively, and we took the connectives of + and = to correspond to ? and >, respectively, then we could say that, given this formal system,

‘S?S>Q.’ In which case the question, ‘Why does S?S>Q ?,’ taken as a question about the formal system, has a simple answer: it’s arbitrary. We could pick any old symbols, so long as we take the symbols to refer successfully. But what the numbers are referring to, it seems to me, are properties of sets, and what we have in mathematics is reality in abstraction, as, e.g., in the conjoining of sets (taken as addition), the removal of entities from sets (taken as substraction), the reiteration of a given set by a specified multiple (as in multiplication), and the separation of entities into new sets as specified by new classes (as in division), et al.

Of course, that last part is sketchy, my rendering some math talk into set talk, and I really don’t know enough set theory or philosophy of mathematics to give a better account now. But my general view of mathematics is that math talk is just set talk and set talk is just the fancy formalizing of everyday ostension and indexing.

Hi Billie Pritchett,

Thanks for the comment!!!

Actually, I am trying to ask a third question ‘what is the proof that 1+1=2; is it empirical or not?’ You suggest that ‘+’ reduces to conjunction and set theory, but this doesn’t really answer the question that I am asking; or if it does it boils down to ‘the evidence is empirical’. My claim is that there is no way for us to distinguish which of thetwo answers is correct…

It seems it could be answered by appeal to someone who understands math proofs with a math proof, and for someone who doesn’t understand formal proofs but who would understand empirical appeals with an empirical appeal. The appeals don’t look as though they would have any deep consequences for mathematics, either, especially if it is the case that math is reducible to set theory and set theory to everyday indexing. What do you think?

The qestion is one of whether or not the math truths depend on experience or not. What is their ultimate justification? Is it something that we generalize from experience or something that reason shows us independantly of experience? My suggestion is that there is no way to answer this question….

[…] entity we have eventually discovered a plausible candidate for a natural explanation. I have also argued that if anything like evolution turns out to be true then we cannot appeal to intuition as evidence […]

So, it has been years since the last post has been made here, but perhaps someoe is still monitoring this page… Here you are, then:

Richard, you said:

“Actually, I am trying to ask a third question ‘what is the proof that 1+1=2; is it empirical or not?”

Does it aid our discussion to note that an “empirical” proof (which, presumably, refers to a thing known by way of experience) itself requires the proposition (i.e., 1 + 1 = 2) that you want to have proven?

What I mean to say is this: At the very least, our notion of experience requires two things. Those things are: a subject/experiencer and an object to be experienced. (I will disregard the possibilities of other necessary factors for an experience – e.g., other people, more than one distinct object of experience, more than one moment of experience, etc.) This much seems necessary: that experience requires two distinct entities. Together they form the experience. If it were the case that either of these two were not existant, then an experience would not occur. Therefore, you have built into your notion of experience the concepts of a thing and another distinct thing which, taken together, form a whole.

So, if we wished, we may substitue our factors here with symbols. The “subject” may – taken as a distinct entity – be referred to as “1.” The same would go for the distinct “object.” Finally, the experience at large could be symbolized as “2” (i.e., the prescence of both factors.)

Now, even if it were necessary to prove empirically the proof of contingent/factual objects’ relationships to one another, we seem to have the assumption of 1 + 1 = 2 already found within our concept of experience. Empirical proof of anything requires the assumption of this propositon.

So, can we prove something which is already assumed within our method of proof?

1+1=2 is true in the special case of mathematics. 1+1=2 is not always true in other cases.

I think the question begs investigation, because science likes to extrapolate the mathematical principal to non-mathematical applicaiton, through theory.

It does work most of the time, but still no one knows what is north of the north pole (as Stephen Hawking Asks).

We never will know either, because our point of reference is always 1,1,1.

This was beautiful to read.

I think the neo-Wittgensteinian is the correct one: 1 + 1 = 2 is necessarily true in the same way that it’s necessarily true that no bachelor is married and no squares are spherical. It’s guaranteed by the meaning of the terms involved.

In any circumstance in which 1 + 1 wasn’t 2, such as one drop of water + one drop of water = one bigger drop of water, we’d have ascribe it to either (a) our failure to apply the concepts correctly or (b) the failure of the concepts to apply to certain objects. As I understand it, whether the concept of quantity applies to a given thing is the empirical question.

1+1=2 because of language. Maths was invented by humans. Humans decided that a single object would be called one and if you had two singles objects, you could put them together and that would be called two. Without humans there would be no human languages and therefore no maths.

i think I would agree with James’s approach : a notion or suggestion based totally on common experience (and common interpretation of this experience) should be refuted or proved or tested by means of common experience – which, of course, is a limited, albeit functional, approach. We can be sure that “one apple plus one apple makes two apples”, but there is no way to determine what “1 + 1” equals, since numbers are out of conception, if not through an empirical intermediate / predicate.

[…] Why does 1+1=2? […]

1 + 1 = 2 is a belief.

This is based on our tendency to see separate things as one. Without this tendency our perceptions disintegrate. Such beliefs allow us to use language and one might argue this leads to consciousness.

There are no ultimate truths.