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# 2+2 = 4

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1. Is this called a Priori?

If so what is the definition of a Priori?
2. (Original post by darkenergy)
Is this called a Priori?

If so what is the definition of a Priori?
Yes, that's a priori.

A priori is information (for want of a better word) which comes from using a rational process. 2 + 2 = 4 can be proven rationally.

A posteriori is information which comes from our senses. It's less reliable in many ways. Believing that there's a God is a posteriori.

I think that's right.
3. 2+2=4 isn't true a priori. A priori means that it is an absolute and basic assumption, based on hypothesis or theory. An a priori judgment is true before or regardless of the facts- here an a priori judgment is the basic hypothesis for maths, is the facts of mathematics. All of number is based on the a priori definitions of 0, 1 and 2. 2-1-1=0 is true a priori. Every other number theory [including 2+2=4] is true a posteriori from that statement.
4. (Original post by Weejimmie)
2+2=4 isn't true a priori. A priori means that it is an absolute and basic assumption, based on hypothesis or theory. An a priori judgment is true before or regardless of the facts- here an a priori judgment is the basic hypothesis for maths, is the facts of mathematics. All of number is based on the a priori definitions of 0, 1 and 2. 2-1-1=0 is true a priori. Every other number theory [including 2+2=4] is true a posteriori from that statement.
Is that not nit-picking?
5. what are axioms then??
6. (Original post by Weejimmie)
2+2=4 isn't true a priori. A priori means that it is an absolute and basic assumption, based on hypothesis or theory. An a priori judgment is true before or regardless of the facts- here an a priori judgment is the basic hypothesis for maths, is the facts of mathematics. All of number is based on the a priori definitions of 0, 1 and 2. 2-1-1=0 is true a priori. Every other number theory [including 2+2=4] is true a posteriori from that statement.
what are the a priori definitions of 2?
7. (Original post by darkenergy)
what are axioms then??
in maths they're things which you accept to be true without proving them. (This definition is rigorous enough for a mathematician but a philosopher would probably find some uber rigorous way of putting it).

example "1 exists"

this axiom is part of a bunch of axioms called peano's axioms (sp?)
8. (Original post by englishstudent)
Is that not nit-picking?
No: number-picking. Intelligent and dedicated philosophers and mathematicians have tried to prove that 1+1=2 without an a priori definition of 1. None of them have succeeded.
An axiom is an a priori assumption. Every field of mathematics and philosophy rests on a number of axioms. One of the aims of many of the people studying these subjects is to reduce the number of axioms to the smallest possible number and to establish that these axioms are mutually supportive. Non-Euclidean geometry shocked people so much because it meant that one of the basic axioms of geometry- that governing parallel lines- had been shown not to be necessarily unchangeable after it had not been questioned for more than two thousand years.
We don't need an a priori definition of 2. The a posteriori definition of 2 is 1+1.
9. Evenin' Philosophers

Was (briefly - it got cut off by Oxbridge panic!) discussing this in another forum t'other day. If no-one minds, I might re-post my answer from there, to save me typing it again.

*goes to find it*

ZarathustraX
10. Ok got it - this was my original post:

(Original post by Juxtapiped)
I was asked of 2+2=4 was 'a priori' (meaning, knowable without facts/evidence/experience). I said it wasn't for some odd reason, and said "ahh its a special kind of...." to which she just said "it is a priori - I think you have your terminology confused".
(Original post by NeuroticSurgeon)
About 2 + 2 = 4 being a priori couldn't that be debatable? I mean I know some philosophers have argued that 2 + 2=4 ISN'T a priori....old philosophers though. Can't remember who it is...(and I might be way off target here). Although the general consensus now is that 2+2=4 is by definition true.
(Original post by Juxtapiped)
I think I tried to argue it from what Immanuel Kant said, and compared it to cause and effect although being universally accepted as true, is not a priori because the 'effect' is not part of the definition of the 'cause' and thus it isnt self-explanatory - or something like the 4 isnt part of the definition of the two 2's, lol - something crazy that was quickly interrupted...
NeuroticSurgeon, you're right it must be possible (though perhaps not successful) to argue that 2+2=4 isn't a priori, as it's pretty much the only line to take on it if you want to maintain absolute empiricism and deny the possibility of any a priori truth. I think it's something along the lines of: we gain the knowledge that 2+2=4 by encountering repeated instances in which 2 pairs of things joined together make four individual things, etc. Then the formula 2+2=4 is abstracted from this and turned into a generalisation. However, it's difficult to see how this works as it's not technically an inductive generalisation, since once you know that 2+2=4 no further repetition in experience makes the truth any more probable, which is the case with inductive knowledge.
("2+2=4 is by definition true" : doesn't that make it analytic, not a priori?)

Juxtapiped, a priori can be taken also to mean any judgement that is logically independent of experience, ie.does not entail any empirical statements, in which case maths is defo a priori - much easier definition to work with.
When you say "the 'effect' is not part of the definition of the 'cause' and thus it isnt self-explanatory" are you talking about Kant's 'predicate-not-contained-within-the-subject' definition of a synthetic statement? Just I've never heard it put that way before...if it's something else please explain and enlighten

ZarathustraX (talking rubbish probably, but trying to draw attention away from paranoia and panic!)

Righto then, someone make me defend that (I probably can't!)

ZarathustraX
11. (Original post by darkenergy)
Is this called a Priori?

If so what is the definition of a Priori?

I fear that everyone has made this much more difficult than it need be. It's apparent that the term 'a priori' admits of various definitions: I shan't use it in the pre-Kantian sense of demonstratio propter quid, but in the Kantian sense of 'independent of all particular experiences'. '2 + 2 = 4', strictly speaking, is a synthetic a priori proposition. The concept of '4' does not include the concept of '2' or the concept of '+', therefore it cannot be analytic; and the proposition is universally true, therefore it cannot be a posteriori. An analytic a priori proposition would be something of the form 'all squares have four sides' (universal and tautological). A synthetic a posteriori proposition would be something like 'John is reading a book': nothing about this act is necessary, and it is easily discoverable through empirical observation.
12. (Original post by svidrigailov)
'2 + 2 = 4', strictly speaking, is a synthetic a priori proposition. The concept of '4' does not include the concept of '2' or the concept of '+', therefore it cannot be analytic; and the proposition is universally true, therefore it cannot be a posteriori.
Sorry, but what is "a synthetic a priori proposition"? I would say- and I think most people who study maths would say- 2+2=4 is an aposteriori proposition. Only the axioms of mathematics are a priori, and there should be as few axioms as possible. The concept 2 is a posteriori [defined as 1+1] and 4 in turn is defined as 2+2. It is only universally true because it is defined in a way that makes it universally true.
13. (Original post by Weejimmie)
Sorry, but what is "a synthetic a priori proposition"? I would say- and I think most people who study maths would say- is an aposteriori proposition.
Weejimmie, re-read the following for a summary of a priori:

(Original post by Zarathustra)
a priori can be taken also to mean any judgement that is logically independent of experience, ie.does not entail any empirical statements, in which case maths is defo a priori
(Original post by svidrigailov)
the Kantian sense of 'independent of all particular experiences'
The proposition 2+2=4 does not logically entail any empirical (or a posteriori) statements.
A priori judgements may have kind of dependence on experience insofar as they may be formed as a result of certain experiences (we see instances of couples and the fact that their combination makes four particulars, but we then reach a point where we can abstract from this the general notion that 2+2=4, without reference to particulars such as ‘two coins’, and realise the necessity of its truth) but they are not logically dependent on experience as they would be equally true in, for example, a world with no countable objects. They are strictly universal & necessary, and their truth can be proved by reason alone.
"2+2=4" is thus a priori because it does not entail any factual judgements regarding what is or is not the case at a certain time.

ZarathustraX
14. (Original post by Zarathustra)
A priori judgements may have kind of dependence on experience insofar as they may be formed as a result of certain experiences (we see instances of couples and the fact that their combination makes four particulars, but we then reach a point where we can abstract from this the general notion that 2+2=4, without reference to particulars such as ‘two coins’, and realise the necessity of its truth) but they are not logically dependent on experience as they would be equally true in, for example, a world with no countable objects. They are strictly universal & necessary, and their truth can be proved by reason alone.
"2+2=4" is thus a priori because it does not entail any factual judgements regarding what is or is not the case at a certain time.

ZarathustraX
Certainly: however, I think that the only a priori statements we need in number are 1-1=0 and 1+1=2. We can abstractly construct the whole number system from those two axioms, so 2+2=4 is a posteriori.
15. (Original post by Weejimmie)
Certainly: however, I think that the only a priori statements we need in number are 1-1=0 and 1+1=2. We can abstractly construct the whole number system from those two axioms, so 2+2=4 is a posteriori.
But if you can construct the entire number system abstractly, through the use of reason alone / without reference to experience, from the a priori statement that 1-1=0, then any statement within mathematics would be logically independent of experience (ie. it would not entail any empirical judgements) and would thus be a priori by virtue of the method of its derivation and the non-empirical nature of its judgements. So, whilst I agree with you that 1-1=0 may the only a priori judgement needed in maths, I wouldn't say that this means the rest of mathematics is a posteriori.

Make sense

ZarathustraX

16. I think this argument is becoming quite repetitious.

1 - 1 = 0 and 1 + 1 = 2, strictly speaking, are not axioms, as they do not have universal applicability, but apply only to particular experience. The axioms, that is to say, the fundamental laws of arithmetic are:

The commutative laws of addition and multiplication:

1) a + b = b + a
2) ab = ba

3) a + (b + c) = (a + b) + c

The associative law of multiplication:

4) a(bc) = (ab)c

The distributive law:

5) a(b + c) = ab + ac

If you wish to extend the domain of positive integers by introducing zero, you'd have to add

a + 0 = a
a.0 = 0
a - a = 0.

For 2 + 2 = 4 to be an a posteriori proposition, as Zarathustra has pointed out, its truth would have to be derived from experience, and not pure reason. Because it is such a small number, it may very well be that you have witnessed an instantiation of this, that is to say, two objects added to two objects, from which you derive the concept of their 'fourness': but in this case appearance is misleading. 2 + 2 = 4 is true for the same reason that 1,237,486 + 2,354,687 = 3,592,173 is true. I have never seen an instantiation of the latter, i. e., I have never counted 1,237,486 individual objects, and added them to 2,354,687 objects that I have counted, and counted the resulting group of objects to obtain the result 3,592,173; and yet I know it is true: its truth is confirmed not by the evidence of experience, but by the operations of reason.
17. Sorry, Svidrigailov, I don't agree. It's a priori propositions that are derived from observation and experience. A posteriori propositio0ns are derived by reason from the a priori propositions.
The fundamental laws of arithmetic can be deduced by observation following from 1-1=0 and 1+1=2. The same is true of every other possible arithmetical sum. You can logically deduce them from those axioms, which surely makes them a posteriori. As you say, 2+2=4 probably will be deduced by observation, but it does not have to be, as the other two must be.
Anyway, as Bertrand Russell thouight his mind was never as good after writing Principia Mathematica, and my mind is nothing like as good as Russell's to begin with, I will withdraw from the topic before i damage it permanently.
18. (Original post by Weejimmie)
Sorry, Svidrigailov, I don't agree. It's a priori propositions that are derived from observation and experience. A posteriori propositio0ns are derived by reason from the a priori propositions.
The fundamental laws of arithmetic can be deduced by observation following from 1-1=0 and 1+1=2. The same is true of every other possible arithmetical sum. You can logically deduce them from those axioms, which surely makes them a posteriori. As you say, 2+2=4 probably will be deduced by observation, but it does not have to be, as the other two must be.
Anyway, as Bertrand Russell thouight his mind was never as good after writing Principia Mathematica, and my mind is nothing like as good as Russell's to begin with, I will withdraw from the topic before i damage it permanently.

Although you have withdrawn, I shall say something nonetheless.

The definition of a particular word or phrase is one of those few instances in philosophy in which one can acknowledge the authority of a particular philosopher or thinker, without leaving oneself open to the charge of fallacious reasoning. “A posteriori” and “ a priori” were used extensively in philosophy from the fourteenth century; the former was taken to mean an argument “from effects to causes”, and the latter, “from causes to effects”. Kant instituted what he called a ‘Copernican revolution’ in philosophical thought, and defined “a priori” and “a posteriori” in such a way as to be considered definitive by all later philosophers and logicians. The Encyclopaedia Britannica says of “a priori”:

"in Western philosophy since the time of Immanuel Kant, knowledge that is independent of all particular experiences, as opposed to a posteriori knowledge, which derives from experience alone."

Kant himself, moreover, in the preface to the Critique of Pure Reason said:

''Such universal cognitions, which at the same time have the character of inner necessity, must be clear and certain for themselves, independently of experience; hence one calls them a priori cognitions; whereas that which is borrowed from experience is, as it is put, cognized only a posteriori, or empirically."

And:

“We will understand those by a priori cognitions not those that occur independently of this or that experience, but rather those that occur absolutely independently of all experience. Opposed to them are empirical cognitions, or those that are possible only a posteriori, i. e., through experience”.

Kant moreover applies certain criteria to the recognition of knowledge as “a priori” or “a posteriori”, the most pertinent of which in this argument is:

“Experience never gives its judgments true or strict but only assumed and comparative universality (through induction), so properly it must be said: as far as we have yet perceived, there is no exception to this or that rule. Thus if a judgment is thought in strict universality, i. e., in such a way that no exceptional at all is allowed to be possible, then it is not derived from experience, but is rather valid absolutely a priori”.

Our mathematical propositions, which we therefore hold to be absolutely true, and not only true as far as we can tell, no instance of their being false having been found by experience, we therefore say are “a priori”. It should be pointed out that a chain of propositions, each built the one upon other, derived solely by means of reason, nonetheless is considered solely “a priori”; it being not the place of a proposition in a sequence, but the form of its cognition, that determines its status. As Bertrand Russell said "All pure mathematics is a priori, like logic".
19. Thanks, Svidrigailov. You live and learn. I'd always thought of a priori as meaning "from the start"- axiomatic, and a posteriori as meaning "afterwards", deducible from a priori statements. Again, I thought that the number of apriori statements we made should be like azioms, as few as possible. I'd also always assumed the opposite of Kant: that a priori propositions must be apparently true at a very high level, from experience, but that a posteriori propositions must follow completely logically from them. If they coincided with experience that only helped to confirm the a priori hypotheses. Probably this misuse came because i first came across the terms in connection with the introduction of non-Euclidean geometry. I am even more determined to stay well away now.
20. (Original post by fishpaste)
what are the a priori definitions of 2?
2 = 1 + 1.
is similar to
12 = 2 + 2 + 2 + 2 + 2 + 2 = (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1)

(Original post by fishpaste)
in maths they're things which you accept to be true without proving them. (This definition is rigorous enough for a mathematician but a philosopher would probably find some uber rigorous way of putting it).

example "1 exists"

this axiom is part of a bunch of axioms called peano's axioms (sp?)
You're right. A philosopher would not accept that "they're things which you accept to be true without proving them" to be the definition of axiom. An axiom is a self-evident principle or one that is accepted as true without proof as the basis for argument; in other words, it's a postulate. The two definitions (of maths and of philosophy) seem quite similar, but look closely, they're not in a sense.

1 + 1 = 2 is not an axiom. You can prove it. There's a mathematical proof (rather than philosophical proof, because the philosophical proof deals with metaphysics too much in explaining why 1 + 1 = 2)

Peano's explanation:

1. 1 is in N

2. there exists a succesor function s:N-->N

3. s(n) is not equal to 1 for all n in N

4. the succesor function is injective

5. If A is a subset of N and 1 is in A and if n is a member of A implies s(n) is a member of A then A = N.

let s(1) = 2

let (n + 1) = s(n)

therefore 1 + 1 = s(1)

therefore 1 + 1 = 2

Another proof:

1) x+0=x
2) x+y'=(x+y)'

(2) says x+the successor of y = the successor of (x+y) -- the ' is the successor function.

By convention, 0=0, 0'=1, 0''=2, and so on (thus we only need two symbols, '0' and ''', for all numbers). For 1+1=2 we have

0'+0'=(0'+0)' (by 2)
=(0')' (by 1)
=0'' (by simplification -- eliminating parentheses)

Therefore 1+1=2

There's also a proven theorem in Principia Mathematica, Whitehead and Russell, (1910), page 62, Volume 2.

Updated: February 24, 2005
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