It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn’t make it 5 just by assuming differently.
“2 + 2 = 4” is not the same type of statement as, say, “sugar dissolves in water”. The statement “sugar dissolves in water” refers to a fact about the world; there are experiments we can perform that would verify that statement.
The statement “2 + 2 = 4”, on the other hand, doesn’t refer to a fact about the world; it refers to a truth-preserving transformation that can be applied to facts about the world. It allows us to transform “I have 2 + 2 apples” into “I have 4 apples”, and “he is 4 years old” to “he is 2 + 2 years old”, and so on.
What do the symbols “2”, “4“, and “+” mean here? Well, they mean a different thing in each context. In one context, “2” means “two apples”, “4“ means “four apples”, and “X + Y” means “X apples, and also Y separate apples”. In the other, “2” means “two years earlier”, “4″ means “four years earlier”, and “X + Y” means “X years earlier than the time that was Y years earlier”.
Why do we use the same symbols with different meanings? Because there happens to be a set of truth-preserving transformations—the ring axioms—that can be applied to all of these different meanings. Since they obey the same axioms, they also obey the transformation “2 + 2 = 4”, which is just a composite of axioms.
Come to think of it, apples don’t actually satisfy the ring axioms. In particular, if you have at least one apple, there is no number of apples I can give you such that you no longer have any apples.
In fancy math-talk, we can say apples are a semimodule over the semiring of natural numbers.
You can add two bunches of apples through the well-known “glomming-together” operation.
You can multiply a bunch of apples by any natural number.
Multiplication distributes over both natural-number addition and glomming-together.
Multiplication-of-apples is associative with multiplication-of-numbers.
1 is an identity with regard to multiplication-of-apples.
You could quibble that there is a finite supply of apples out there, so that (3 apples) + (all the apples) is undefined, but this model ought to work well enough for small collections of apples.
Nor is it obvious how multiplication of apples should work. Apples might be considered an infinite cyclic abelian monoid, if you like, but it’s beside the point—the point is that once you know what axioms they satisfy, you now know a whole bunch of stuff.
Well, if you have a row of 3 apples, and you get another three rows, you’ll have 9 apples. But multiplying 3 apples by 3 apples would result in 9 apples^2; and I don’t know what those look like.
Does an apple composed of antimatter still count as an apple? If I add that to the original apple, I get a big explosion and lots of energy flying around, but neither apple actually remains afterwards.
Well, strictly speaking we don’t directly assume that 2+2=4. We have some basic assumptions about counting and addition, and it follows from these that 2+2=4. But that doesn’t really avoid the objection, it just moves it down the chain.
Can I change these assumptions? Well, firstly it bears saying that if I do, I’m not really talking about counting or addition any more, in the same way that if I define “beaver” to mean “300 ton sub-Saharan lizard”, I’m not really talking about beavers.
So suppose I change my assumptions about counting and addition such that it came out that 2+2=5. Would that mean that two apples added to two apples made five apples? Obviously not. It would mean that two apples added to two apples made five apples, where the starred words refer to altered concepts.
Anyway, I prefer to see 2+2=4 as deriving from set theory, rather than arithmetic. Set theory has its formal rules, and some version of 2+2=4 is one of them.
The question is, do we find things in our world that can be usefully modelled by set theory? We can. For a start, there seem to be many objects that have persistence—cups, trees and planets don’t generally split or multiply in the course of a conversation. Also, we can usefully group objects together by their properties, and it is often useful to ignore their differences. Two similar looking trees are likely to do similar kinds of things in similar situations (ie induction over classes of objects is not completely useless).
So two cups of water plus two cups of sugar makes four cups—as long as we’re interested in cups, not in the difference between sugar and water. Two elms and two oaks make four trees—as long as we’re interest in trees in general, and as long as we aren’t thinking on the scale of decades, and as long there isn’t a guy with a chainsaw and an itchy trigger finger.
In fact, set theory is so useful about so many things in the world, that we abstract it to a universal truth − 2+2=4 in so many different circumstances, that simply 2+2=4.
So we can’t make two apples plus two apples equal five apples (at least not in a few seconds) because apples, in the way that we use the term, and on the time scales that we use the term, are objects that obey the axioms of set theory.
All the premises necessary to prove that 2+2=4 can be found in the definitions of 2, 4, + and =. Add in some definitions from set theory, and you get sizeof(X)=2 && sizeof(Y)=2 and disjoint(X,Y) |- sizeof(X u Y)=4. The trickier assumptions are in converting from operations on apples to operations on sets. This isomorphism is entailed by the counting algorithm; if there’s no person around to count them, then when we said “there are two apples” we were talking about a counterfactual in which something did run the counting algorithm. (Note that the abstract counting algorithm assumes a perfectly-reliable tagging of counted and not-counted-yet objects, and a perfectly reliable incrementing number; a count made by a human has neither of these things, so sometimes counted_as(25,000)+counted_as(25,000gp) = counted_as(50,001).)
This doesn’t seem directly relevant to the meditation. Are you saying that, yes, it is an assumption, because it’s an argument from definition? Because I guess that would be responsive, but the more interesting question is whether or not those definitions of 2, 4, +, and = can be known to correspond to the external world through more than just our assumptions.
We might mean many things by “2 + 2 = 4”. In PA:
“PA |- SS0 + SS0 = SSSS0″, and so by soundness “PA |= SS0 + SS0 = SSSS0”
In that sense, it is a logical truism independent of people counting apples. Of course, this is clearly not what most people mean by “2+2=4″, if for no other reason than people did number theory before Peano.
When applied to apples, “2 + 2 = 4” probably is meant as:
“apples + the world |= 2 apples + 2 apples = 4 apples”.
the truth of which depends on the nature of “the world”. It seems to be a correct statement about apples. Technically I have not checked this property of apples recently, but when I consider placing 2 apples on a table, and then 2 more, I think I can remove 4 apples and have none left. It seems that if I require 4 apples, it suffices to find 2 and then 2 more. This is also true of envelopes, paperclips, M&M’s and other objects I use. So I generalise a law like behaviour of the world that “2 things + 2 things makes 4 things, for ordinary sorts of things (eg. apples)”.
At some level, this is part of why I care about things that PA entails, rather than an arbitrary symbol game; it seems that PA is a logical structure that extracts lawlike behaviour of the world. If I assumed a different system, I might get “2+2=5”, but then I don’t think the system would correspond to the behaviours of apples and M&M’s that I want to generalise.
(On the other hand, PA clearly isn’t enough; it seems to me that strengthened finite Ramsey is true, but PA doesn’t show it. But then we get into ZFC / second order arithmetic, and then systems at least as strong as PA_ordinal, and still lose because there are no infinite descending chains in the ordinals)
2, +, 2, =, and 4 are just definitions. What they are definitions for depends on the underlying representation (2 might be a definition for S(S(0)) in PA, { {} , {{}} } in ZF set theory, or two apples in school) but what really matters is that there exists a homomorphism between all our representations.
We can convert between any of our representations while preserving the structure of the relationships between the objects in our representations. What we have discovered is not that “2 + 2 = 4” was always true but that any possible equivalent representation is an inherent property of the universe.
“2 + 2 = 5” just lacks a homomorphism to any other useful representations of reality based on our common definitions.
One does not have to directly assume that 2 + 2 = 4. Note that all of the following statements correctly describe first-order logic (given the appropriate definitions of 2, 3, and 4):
So ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ 2 + 2 = 4 (in the context of first-order logic). By the soundness theorem for first-order logic, then,
∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊨ 2 + 2 = 4. (i.e. In all models where “∀x ∀y ((x + y) + 1 = x + (y + 1))” is true, “2 + 2 = 4″ is also true.)
So, yes, an assumption is being made, and that assumption is “∀x ∀y ((x + y) + 1 = x + (y + 1))”. To evaluate the plausibility of that assumption, we now must specify a domain of discourse, the meaning of ‘+’, and the meaning of ‘1’. Eliezer means to (I hope!) specify the natural numbers, addition, and one, respectively. Addition over the natural numbers is itself an abstraction of moving different quantities of discrete non-microscopic objects (like apples) next to each other, and observing the quantity after that. Now we can evaluate the plausibility of “∀x ∀y ((x + y) + 1 = x + (y + 1))”, or, to translate into English, “If you take a group of y objects and move it next to a group of x objects, and then move one more object next to that newly formed group of objects, and then observe the quantity, then you will observe the same quantity as if you take a group of y objects and move 1 more object next to it, and then move that newly formed group of objects next to a group of x objects, and observe the quantity, no matter what the original numbers x and y are.” (So symbolic language is useful after all....) We observe evidence for that statement every day! It is a belief that pays rent in terms of anticipated experience. So, yes, we could assume differently, but we would be doing so in the face of mountains of evidence pointing in the other direction, and with something to protect, that won’t do.
Notes:
How do we know the soundness theorem for first-order logic is true? For one thing, we observe inductive evidence for it all the time (the rules of first-order logic are incredibly intuitive, see page 51 (PDF page 57) of A Primer for Logic and Proof for an enumeration), just as we see inductive evidence for “∀x ∀y ((x + y) + 1 = x + (y + 1))” all the time. We can also use mathematical induction (on the length of a deduction) to prove the soundness theorem, but this probably won’t convince someone who is skeptical of even first-order logic.
It might be helpful to think of (X ⊨ Y) as meaning P(Y|X) = 1.
Yes, we are making an assumption, and yes, (if it is true) it was true well before anyone was around to assume it, and yes, making a different assumption does not change it’s truth value. That’s part of what “assumption” means in this sense.
The fact that if we put any two objects into the same (previously empty) basket as any other two object we will in this basket have four objects is true before we can make any definitions. But the statement 2 + 2 = 4 does not make any sense before we have invented: (a) the numerals 2 and 4, (b) the symbol for addition + and (c) the symbol for equality =. When we have invented meanings for these symbols (symbols as things we use in formal manipulations are quite different from words and were invented quite late, much later than we started to actually solve problems mathematically) we have to show that they correspond to our intuitive meaning of putting things into baskets and counting them, but if they do they will also satisfy, say the Peano axioms for the natural numbers, which are the axioms we tend to start from to prove statements like 2+2=4 or “there are infinitely many prime numbers”.
If we were to assume differently such that 2+2 =5, then our notion of + and = would not correspond to the notions of adding objects to a basket and counting them. This is because we could walk through our proof step by step (as described in this post) to find the first line where we write down something that is not true for our usual notion of adding apples, there we would have an assumption or a rule of inference that was assumed in this new theory but which does not correspond to apple comparison.
When people say “2+2=4”, what do they mean? Well, “=” is a standard symbol in logic (IIRC it can be derived from purely syntactical rules). But 2 and 4 and + aren’t standard; they are defined as part of the model you’re working with. For instance, if your model is boolean algebra, there are no 2 or 4, there are only ‘true’ and ‘false’, and “2+2=4” isn’t valid or invalid, it’s syntactically meaningless.
Depending on the model you choose, the sentence “2+2=4” may be true (as for the Peano integers), or undefined (as for boolean algebra), or even false (exchange the usual meanings of the strings “2“ and “4”). The latter case would be human-perverse but mathematically-sound. But once you choose a a model, every sentence—including “2+2=4”—is either a logical truth (is valid) or it is not.
Now, there are some standard (in the sense of widely used) models where 2 and 4 and + are used as symbols. These include the integers, the real numbers, etc. Usually when people say “2+2=4”, they are thinking of one of these. And in these standard models, “2+2=4″ is indeed a logical truth.
So given a model, we are not assuming 2+2=4. Our choice of a standard model dictates that “2+2=4” is true, without extra assumptions. A different model might dictate that “2+2=4″ is false, or undefined.
But we are relying on our shared assumption about what model we’re talking about—which is a different kind of assumption; it’s not about whether “2+2=4” is true (valid), but about what the string means—how to read it. It’s similar to the assumption that we’re speaking English, rather than a different language in which all the words just happen to mean something else.
My response, before reading the other responses, is that this is not a matter of assumption but of definition; the symbols ‘2’, ‘+’, ‘=’ and ‘4’ have been defined in such a way that 2+2=4 is a true statement. (The important symbol, here, is + in my view: 2, 4 and = are such basic operations that it’s near certain that there would have been some symbol with those meanings. + is pretty basic, but to my mind less basic—it’s not the only way to combine two quantities).
This could be seen as taking the definitions of 2, 4, = and + as premises and the truth of the statement 2+2=4 as a conclusion. The conclusion (2+2=4) follows from the premises whether anyone’s around to postulate the premises or not; that remains true of any logical statement. So, similarly, if all kittens are little, and if all little things are innocent, then it remains true that all kittens are innocent whether anyone has ever considered that chain of logic before or not.
It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equaled two apples before there was anyone to count them, and that we couldn’t make it 5 just by assuming differently.
I think it’s an assumption, depending on what you mean when you say something is an assumption. It seems to us like “2 + 2 = 4”, but if we assumed differently then it would seem like “2 + 2 = 5“. We seem to have justifications to point to to believe that “2 + 2 = 4”, but if we assumed that “2 + 2 = 5” then we would also seem to have justifications for that point of view.
I personally have no problem with this and will continue to follow my current beliefs.
Meditation:
It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn’t make it 5 just by assuming differently.
My two cents:
“2 + 2 = 4” is not the same type of statement as, say, “sugar dissolves in water”. The statement “sugar dissolves in water” refers to a fact about the world; there are experiments we can perform that would verify that statement.
The statement “2 + 2 = 4”, on the other hand, doesn’t refer to a fact about the world; it refers to a truth-preserving transformation that can be applied to facts about the world. It allows us to transform “I have 2 + 2 apples” into “I have 4 apples”, and “he is 4 years old” to “he is 2 + 2 years old”, and so on.
What do the symbols “2”, “4“, and “+” mean here? Well, they mean a different thing in each context. In one context, “2” means “two apples”, “4“ means “four apples”, and “X + Y” means “X apples, and also Y separate apples”. In the other, “2” means “two years earlier”, “4″ means “four years earlier”, and “X + Y” means “X years earlier than the time that was Y years earlier”.
Why do we use the same symbols with different meanings? Because there happens to be a set of truth-preserving transformations—the ring axioms—that can be applied to all of these different meanings. Since they obey the same axioms, they also obey the transformation “2 + 2 = 4”, which is just a composite of axioms.
Come to think of it, apples don’t actually satisfy the ring axioms. In particular, if you have at least one apple, there is no number of apples I can give you such that you no longer have any apples.
In fancy math-talk, we can say apples are a semimodule over the semiring of natural numbers.
You can add two bunches of apples through the well-known “glomming-together” operation.
You can multiply a bunch of apples by any natural number.
Multiplication distributes over both natural-number addition and glomming-together.
Multiplication-of-apples is associative with multiplication-of-numbers.
1 is an identity with regard to multiplication-of-apples.
You could quibble that there is a finite supply of apples out there, so that (3 apples) + (all the apples) is undefined, but this model ought to work well enough for small collections of apples.
Apples (and other finite sets of concrete objects) form a semiring.
Nor is it obvious how multiplication of apples should work. Apples might be considered an infinite cyclic abelian monoid, if you like, but it’s beside the point—the point is that once you know what axioms they satisfy, you now know a whole bunch of stuff.
Well, if you have a row of 3 apples, and you get another three rows, you’ll have 9 apples. But multiplying 3 apples by 3 apples would result in 9 apples^2; and I don’t know what those look like.
Sure there is, as long as you realize that “give” and “take” are the same action. Giving −1 apples is just taking 1 apple.
Number of apples isn’t closed under taking.
If you wish to satisfy the ring axioms from scratch, you must first invent the economy...
Does an apple composed of antimatter still count as an apple? If I add that to the original apple, I get a big explosion and lots of energy flying around, but neither apple actually remains afterwards.
Well, strictly speaking we don’t directly assume that 2+2=4. We have some basic assumptions about counting and addition, and it follows from these that 2+2=4. But that doesn’t really avoid the objection, it just moves it down the chain.
Can I change these assumptions? Well, firstly it bears saying that if I do, I’m not really talking about counting or addition any more, in the same way that if I define “beaver” to mean “300 ton sub-Saharan lizard”, I’m not really talking about beavers.
So suppose I change my assumptions about counting and addition such that it came out that 2+2=5. Would that mean that two apples added to two apples made five apples? Obviously not. It would mean that two apples added to two apples made five apples, where the starred words refer to altered concepts.
That’s what I was trying to say, but I couldn’t find a decent way to put it and gave up.
Not exactly on subject, but the numbers up to 5 seem wired into our brain, so in effect, 2+2=4 before you were old enough to know what “2” was.
Anyway, I prefer to see 2+2=4 as deriving from set theory, rather than arithmetic. Set theory has its formal rules, and some version of 2+2=4 is one of them.
The question is, do we find things in our world that can be usefully modelled by set theory? We can. For a start, there seem to be many objects that have persistence—cups, trees and planets don’t generally split or multiply in the course of a conversation. Also, we can usefully group objects together by their properties, and it is often useful to ignore their differences. Two similar looking trees are likely to do similar kinds of things in similar situations (ie induction over classes of objects is not completely useless).
So two cups of water plus two cups of sugar makes four cups—as long as we’re interested in cups, not in the difference between sugar and water. Two elms and two oaks make four trees—as long as we’re interest in trees in general, and as long as we aren’t thinking on the scale of decades, and as long there isn’t a guy with a chainsaw and an itchy trigger finger.
In fact, set theory is so useful about so many things in the world, that we abstract it to a universal truth − 2+2=4 in so many different circumstances, that simply 2+2=4.
So we can’t make two apples plus two apples equal five apples (at least not in a few seconds) because apples, in the way that we use the term, and on the time scales that we use the term, are objects that obey the axioms of set theory.
All the premises necessary to prove that 2+2=4 can be found in the definitions of 2, 4, + and =. Add in some definitions from set theory, and you get sizeof(X)=2 && sizeof(Y)=2 and disjoint(X,Y) |- sizeof(X u Y)=4. The trickier assumptions are in converting from operations on apples to operations on sets. This isomorphism is entailed by the counting algorithm; if there’s no person around to count them, then when we said “there are two apples” we were talking about a counterfactual in which something did run the counting algorithm. (Note that the abstract counting algorithm assumes a perfectly-reliable tagging of counted and not-counted-yet objects, and a perfectly reliable incrementing number; a count made by a human has neither of these things, so sometimes counted_as(25,000)+counted_as(25,000gp) = counted_as(50,001).)
This doesn’t seem directly relevant to the meditation. Are you saying that, yes, it is an assumption, because it’s an argument from definition? Because I guess that would be responsive, but the more interesting question is whether or not those definitions of 2, 4, +, and = can be known to correspond to the external world through more than just our assumptions.
We might mean many things by “2 + 2 = 4”. In PA: “PA |- SS0 + SS0 = SSSS0″, and so by soundness “PA |= SS0 + SS0 = SSSS0” In that sense, it is a logical truism independent of people counting apples. Of course, this is clearly not what most people mean by “2+2=4″, if for no other reason than people did number theory before Peano.
When applied to apples, “2 + 2 = 4” probably is meant as: “apples + the world |= 2 apples + 2 apples = 4 apples”. the truth of which depends on the nature of “the world”. It seems to be a correct statement about apples. Technically I have not checked this property of apples recently, but when I consider placing 2 apples on a table, and then 2 more, I think I can remove 4 apples and have none left. It seems that if I require 4 apples, it suffices to find 2 and then 2 more. This is also true of envelopes, paperclips, M&M’s and other objects I use. So I generalise a law like behaviour of the world that “2 things + 2 things makes 4 things, for ordinary sorts of things (eg. apples)”.
At some level, this is part of why I care about things that PA entails, rather than an arbitrary symbol game; it seems that PA is a logical structure that extracts lawlike behaviour of the world. If I assumed a different system, I might get “2+2=5”, but then I don’t think the system would correspond to the behaviours of apples and M&M’s that I want to generalise.
(On the other hand, PA clearly isn’t enough; it seems to me that strengthened finite Ramsey is true, but PA doesn’t show it. But then we get into ZFC / second order arithmetic, and then systems at least as strong as PA_ordinal, and still lose because there are no infinite descending chains in the ordinals)
Two cups of water plus two cups of sugar does not equal four cups of sugar water. ;)
2, +, 2, =, and 4 are just definitions. What they are definitions for depends on the underlying representation (2 might be a definition for S(S(0)) in PA, { {} , {{}} } in ZF set theory, or two apples in school) but what really matters is that there exists a homomorphism between all our representations.
%20+%5E{school}%20H_{school}(2%20apples)%20=%20H_{school}(2%20apples%20+%202%20apples))Even better, there’s generally an inverse homomorphism back the real world.
)))%20+%20H_{school}%5E{-1}(%20H_{ZF}(\{%20\{\}%20,%20\{\{\}\}%20\})))We can convert between any of our representations while preserving the structure of the relationships between the objects in our representations. What we have discovered is not that “2 + 2 = 4” was always true but that any possible equivalent representation is an inherent property of the universe.
“2 + 2 = 5” just lacks a homomorphism to any other useful representations of reality based on our common definitions.
This is a tricky one...here’s my attempt:
One does not have to directly assume that 2 + 2 = 4. Note that all of the following statements correctly describe first-order logic (given the appropriate definitions of 2, 3, and 4):
∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ (2 + 1) + 1 = 2 + (1 + 1)
(2 + 1) + 1 = 2 + (1 + 1) ⊢ 3 + 1 = 2 + (1 + 1)
3 + 1 = 2 + (1 + 1) ⊢ 4 = 2 + (1 + 1)
4 = 2 + (1 + 1) ⊢ 4 = 2 + 2
4 = 2 + 2 ⊢ 2 + 2 = 4
So ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊢ 2 + 2 = 4 (in the context of first-order logic). By the soundness theorem for first-order logic, then, ∀x ∀y ((x + y) + 1 = x + (y + 1)) ⊨ 2 + 2 = 4. (i.e. In all models where “∀x ∀y ((x + y) + 1 = x + (y + 1))” is true, “2 + 2 = 4″ is also true.)
So, yes, an assumption is being made, and that assumption is “∀x ∀y ((x + y) + 1 = x + (y + 1))”. To evaluate the plausibility of that assumption, we now must specify a domain of discourse, the meaning of ‘+’, and the meaning of ‘1’. Eliezer means to (I hope!) specify the natural numbers, addition, and one, respectively. Addition over the natural numbers is itself an abstraction of moving different quantities of discrete non-microscopic objects (like apples) next to each other, and observing the quantity after that. Now we can evaluate the plausibility of “∀x ∀y ((x + y) + 1 = x + (y + 1))”, or, to translate into English, “If you take a group of y objects and move it next to a group of x objects, and then move one more object next to that newly formed group of objects, and then observe the quantity, then you will observe the same quantity as if you take a group of y objects and move 1 more object next to it, and then move that newly formed group of objects next to a group of x objects, and observe the quantity, no matter what the original numbers x and y are.” (So symbolic language is useful after all....) We observe evidence for that statement every day! It is a belief that pays rent in terms of anticipated experience. So, yes, we could assume differently, but we would be doing so in the face of mountains of evidence pointing in the other direction, and with something to protect, that won’t do.
Notes:
How do we know the soundness theorem for first-order logic is true? For one thing, we observe inductive evidence for it all the time (the rules of first-order logic are incredibly intuitive, see page 51 (PDF page 57) of A Primer for Logic and Proof for an enumeration), just as we see inductive evidence for “∀x ∀y ((x + y) + 1 = x + (y + 1))” all the time. We can also use mathematical induction (on the length of a deduction) to prove the soundness theorem, but this probably won’t convince someone who is skeptical of even first-order logic.
It might be helpful to think of (X ⊨ Y) as meaning P(Y|X) = 1.
Yes, we are making an assumption, and yes, (if it is true) it was true well before anyone was around to assume it, and yes, making a different assumption does not change it’s truth value. That’s part of what “assumption” means in this sense.
The fact that if we put any two objects into the same (previously empty) basket as any other two object we will in this basket have four objects is true before we can make any definitions. But the statement 2 + 2 = 4 does not make any sense before we have invented: (a) the numerals 2 and 4, (b) the symbol for addition + and (c) the symbol for equality =. When we have invented meanings for these symbols (symbols as things we use in formal manipulations are quite different from words and were invented quite late, much later than we started to actually solve problems mathematically) we have to show that they correspond to our intuitive meaning of putting things into baskets and counting them, but if they do they will also satisfy, say the Peano axioms for the natural numbers, which are the axioms we tend to start from to prove statements like 2+2=4 or “there are infinitely many prime numbers”.
If we were to assume differently such that 2+2 =5, then our notion of + and = would not correspond to the notions of adding objects to a basket and counting them. This is because we could walk through our proof step by step (as described in this post) to find the first line where we write down something that is not true for our usual notion of adding apples, there we would have an assumption or a rule of inference that was assumed in this new theory but which does not correspond to apple comparison.
When people say “2+2=4”, what do they mean? Well, “=” is a standard symbol in logic (IIRC it can be derived from purely syntactical rules). But 2 and 4 and + aren’t standard; they are defined as part of the model you’re working with. For instance, if your model is boolean algebra, there are no 2 or 4, there are only ‘true’ and ‘false’, and “2+2=4” isn’t valid or invalid, it’s syntactically meaningless.
Depending on the model you choose, the sentence “2+2=4” may be true (as for the Peano integers), or undefined (as for boolean algebra), or even false (exchange the usual meanings of the strings “2“ and “4”). The latter case would be human-perverse but mathematically-sound. But once you choose a a model, every sentence—including “2+2=4”—is either a logical truth (is valid) or it is not.
Now, there are some standard (in the sense of widely used) models where 2 and 4 and + are used as symbols. These include the integers, the real numbers, etc. Usually when people say “2+2=4”, they are thinking of one of these. And in these standard models, “2+2=4″ is indeed a logical truth.
So given a model, we are not assuming 2+2=4. Our choice of a standard model dictates that “2+2=4” is true, without extra assumptions. A different model might dictate that “2+2=4″ is false, or undefined.
But we are relying on our shared assumption about what model we’re talking about—which is a different kind of assumption; it’s not about whether “2+2=4” is true (valid), but about what the string means—how to read it. It’s similar to the assumption that we’re speaking English, rather than a different language in which all the words just happen to mean something else.
My response, before reading the other responses, is that this is not a matter of assumption but of definition; the symbols ‘2’, ‘+’, ‘=’ and ‘4’ have been defined in such a way that 2+2=4 is a true statement. (The important symbol, here, is + in my view: 2, 4 and = are such basic operations that it’s near certain that there would have been some symbol with those meanings. + is pretty basic, but to my mind less basic—it’s not the only way to combine two quantities).
This could be seen as taking the definitions of 2, 4, = and + as premises and the truth of the statement 2+2=4 as a conclusion. The conclusion (2+2=4) follows from the premises whether anyone’s around to postulate the premises or not; that remains true of any logical statement. So, similarly, if all kittens are little, and if all little things are innocent, then it remains true that all kittens are innocent whether anyone has ever considered that chain of logic before or not.
I think it’s an assumption, depending on what you mean when you say something is an assumption. It seems to us like “2 + 2 = 4”, but if we assumed differently then it would seem like “2 + 2 = 5“. We seem to have justifications to point to to believe that “2 + 2 = 4”, but if we assumed that “2 + 2 = 5” then we would also seem to have justifications for that point of view.
I personally have no problem with this and will continue to follow my current beliefs.