It is clear that 2 + 2 = 4. It is also clear that applying the successor function on 1 yields the next number, i.e. 2, and this operation can be repeated infinitely. This method can be used to verify the statement 2 + 2 = 4, with symbols +, = properly defined.

What I wish to ask is the following: Is 2 + 2 = 4 true by virtue of itself, or is it true because it can be verified by atomic operations composed of the successor function? This question can be broken down to the following:

Does the number 2 exist strictly in relation to the number 1, or is it independent of 1? That is, do all numbers exist together and independently of each other, and we interpret/learn about them as one of them being the successor of the other (and then create a successor function theory to formalise numbers), OR do we create 2, 3, ... from 1 by application of the successor function?

My point of view is that If we take 1 to exist independently, there is NO reason to not acknowledge any other number's independent existence.

  • 1
    It is true in Peano arithmetic because we can prove it from axioms. – Mauro ALLEGRANZA May 8 '19 at 14:32
  • 2
    Under Platonism, numbers are "independently existing" platonic forms that enter relations with each other, tracing those relations allows to establish truths about them. Under formalism, they are only meaningful as place holders in a relational system that we created for our purposes. We are not going to resolve the perennial dispute about the nature of mathematics here, so what sort of answer are you looking for? – Conifold May 8 '19 at 17:52

In Peano arithmetic 2 is defined as the successor of 1 (in symbols : s(1)) and 1 in turn is s(0).

Thus :

2 = s(s(0)).

In the same way : 4 = s(3) = s(s(s(s(0)))).

To prove the equation : 2 + 2 = 4 amounts to prove :

s(s(0)) + s(s(0)) = s(s(s(s(0)))).

Repeated application of the axiom : n + s(m) = s(n + m) will produce the desired result.

The "number sequence" : 0, 1, 2, ... is characterized exactly by the fact that each number (except 0) is the successor of the previous one.

This does not necessarily conflicts with the "philosophical" assumption that (natural) numbers exist all together.

IMO, to state that each number exists independently from the others can be more difficult to elucidate.

  • I think humans have been adding numbers long before Peano was born. What would be the explanation for that time period? Was there no proof back then? – Cell May 8 '19 at 15:26
  • @Cell - yes, obviously the "origin" is with counting. We count two eggs and put them on the floor; then we count two new eggs and we put on the floor near the previous ones. Then we count the total and we find that they are four. What are two and four ? Names associated to fingers, maybe... – Mauro ALLEGRANZA May 8 '19 at 15:29
  • It is obvious that eggs (and fingers) exist in reality and independently from our "counting process". The issue is : counting numbers (as abstarct objects, i.e. abstracted from the real process of counting eggs and fingers) what are they ? In what sense they exist independently from eggs ? The "best" explanation is that they exist as elements of the number sequence. – Mauro ALLEGRANZA May 8 '19 at 15:30
  • Hmm I never used the word counting it was your decision to go that far back. But if 2 eggs and 2 eggs is 4 eggs. And 2 fingers and 2 fingers is 4 fingers. Why is it not sufficient to say we can generalize this by removing the units and keeping the magnitudes for abstract objects? – Cell May 8 '19 at 16:11
  • @Cell We can. If we can name 2 as 1+1, 3 as 2+1, and 4 as 3+1. and agree that generally n+(m+1)=(n+1)+m, then we can show 2+2=4. via 2+2 = 2+(1+1) = (2+1)+1 =3+1= 4. – Graham Kemp May 9 '19 at 3:52

It may be better to start with 0 as the pre-eminent number rather than 1 and leave 1 undefined except to the extent the successor function needed it for incrementing. This would allow one to conclude that these numbers exist as a set within logic. The successor function would define these numbers existing as members of the set. As members of a set they are "distinct objects that make up that set" (Wikipedia) and hence exist independently of each other.

However, what the OP appears to desire is not to claim that any of these numbers exist outside of perhaps a pre-eminent number (0 or 1). This may be possible. Wittgenstein objected to the existence of these numbers, including the pre-eminent numbers. He would provide an example of how this might be done.

G. E. M. Anscombe describes Wittgenstein's position in comparison to Frege and Russell as follows: (page 126)

For Frege and Russell (natural) number was not a formal concept, but a genuine concept that applied to some but not all objects (Frege) or to some but not all classes of classes (Russell); those objects, or classes, to which the concept number applied were picked out from others of their logical type as being 0 and the successors of 0.

So it is not necessary to consider numbers as genuine concepts, that is, as something more than a formal concept in logic.

If one takes an approach like Wittgenstein's one may be able to avoid the need for these numbers existing except as pointing to "which term it is, which performance of the generating operation the term results from" (page 126).

For more detail on how Wittgenstein viewed numbers through their use as the exponents in any formal series, see pmfcolling's question: What does Wittgenstein mean when he says "there are no numbers in logic"?, the answers provided and Wittgenstein's Tractatus Logico-Philosophus 6.01 and following.

Anscombe, G. E. M. An Introduction to Wittgenstein's Tratatus. 1971. St. Augustine's Press.

Wikipedia contributors. (2019, April 19). Element (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 14:37, May 8, 2019, from https://en.wikipedia.org/w/index.php?title=Element_(mathematics)&oldid=893194907

  • How different is the argument that There are no pre-eminent numbers from the argument that All numbers are pre-eminent? – Ajax May 8 '19 at 19:02
  • @Ajax The pre-eminent numbers would be the initial number (0 or 1) that the successor function used as a starting point and for the increment. These are like axioms. From them one gets all the others. You could call all of them pre-eminent, but then you would need some other way to describe 0 or 1. Wittgenstein's point was that not even these pre-eminent numbers, 0 or 1, were necessary. They were also defined as a formal rather than a genuine concept. As numbers they did not exist as a set or class in logic. – Frank Hubeny May 8 '19 at 20:47
  • I am sorry, I am quite uneducated in this area, and cannot understand that If I decide to designate all numbers as pre-eminent, why do I need to describe 0 or 1? All numbers acquire equal significance. Either every number should be pre-eminent or none. If the number 1 is pre-eminent, then WHATEVER characterizes 1, THAT also logically implies that all other numbers simultaneously and pre-eminently exist (we Get 2, Repeat, Get 3, and so on), otherwise 1 loses its meaning, and it would be senseless to think of the number 1. Characterization is in the difference. What is 1 if there isn't 2? – Ajax May 9 '19 at 7:50
  • @Ajax Think of pre-eminent as numbers that stand out. The reason for 0 to stand out is because one needs some place for the successor function to start. That might as well be 0. The increment of 1 used by the successor function might be a reason for it to stand out. So one needs these to exist in some way first and then one can get all the others. – Frank Hubeny May 9 '19 at 10:30
  • That I understand. What I cannot reason is why does 0 need any description if I can brand every number including 0 as pre-eminent? It is senseless to think of 0 without thinking about a non-zero entity. What really defines a number in our minds is not its face value (1 represents one unit), but WHAT it is NOT. That 1 is not 2 and is not 0. That it differs. That fully describes the number 1 in our minds. 1 means nothing if there isn't 2 to contrast it (i.e. we cannot avoid thinking of 2 when thinking of 1; it helps us to mentally define it). – Ajax May 9 '19 at 18:50

Its true because of what we think of a single unit as. For example one apple or one stick. When you put them next to each other they preserve their identity or individuality. This is not true for all things. If you place a drop of water closer and closer to another, they eventually cohere into one drop of water. The point I'm making is that there are many other kinds of arithmetic. There are arithmetics where 1+1=0, for example.

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