As we know a mathematical operation is a function. And a function is a special type of relation. Ultimately a relation is a set of ordered pairs. For example, what is addition of natural numbers? At the most abstract level, addition is a functional relation=correspondence between an ordered set (a,b) of two natural numbers and another number c. So in a rigorous way addition is just a set containing elements of the form ((a,b),c) for example: +={....; ((1,2),3); ((2,3),5); ((7,0),7),...} Here we can say that c is the sum of a and b in this order (the order should not matter because of commutativity )

So why it is addition viewed by almost everyone as a "process" with a "result"? What in the world is a "process" in mathematics? Can be the notion of "mathematical process" be rigorously defined? I don;t see how... And if we want, we could even critique the writing "1+2=3"?? We know very well what "3" is, but what is "1+2"?? What type of mathematical object is that? How can a mathematical object be equal to a combination" of two other objects? Oh, we say 1+2 is just a "symbolic expression" or just another "name" for 3... i find this a bit unsatisfying from a highly rigorous point of view. Even some mathematicians when seeing "1+2=3" think like this: well we have a process of adding 1 and 2 and we get three... i don;t like this interpretation. At all! Others even say "that is the performing of a computation"...What the heck? So on the left side we have an "unperformed computation" and on the right side a result?? What's a performed and unperformed computation?? How can be something unperformed equal to other something?? How can a single object be equal to a combination of other two objects?? As in 1+2=3 And then, shouldn't 1+2=3 be different than "1+2=3+0"?? In the latter we have an equality of two "combinations", in the first we have an equality between a single combination and its result... from a highly rigorous point of view the tqo equal signs "Act" a bit different.

So if we want complete rigor in math, statements as "1+2=3" lack it a bit... because "1+2" and "3" should be of different types, so how can they be equal?

So my questions are: 1) Speaking from a purely SYNTACTICAL point of view, how should we rigorously define what a "combination" of two mathematical objects is, for example two numbers? If 1,2,3,... are a "type" of math objects, shouldn't "1+2" be of another type??

2) How can two things be equal if they are of different types? For example in 1+2=3, LHS is a "combination" of two numbers and RHS is a single number... how can they be treated as equal, having different types?

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    A "process" in math can be modelled as a computation and we have Computability theory. Dec 30, 2018 at 15:39
  • 1
    Why ""1+2" and "3" should be of different types" ? They are arithmetical terms, i.e. "names" for number. Dec 30, 2018 at 15:41
  • @MauroALLEGRANZA I think here "1+2"is the (or a) "process" statement of "3".
    – christo183
    Dec 30, 2018 at 16:08
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    See Girard's book "Proofs and Types" paultaylor.eu/stable/prot.pdf Page 1, "Section 1.1 Sense and denotation in logic", discusses >>exactly<< your example (including whether we should consider the operation as a graph, in the traditional Dirichlet sense, or as a process).
    – user19423
    Dec 30, 2018 at 17:03
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    " Ultimately a relation is a set of ordered pairs." No, this is not true. A relation is modeled within set theory as a set of ordered pairs. You are confusing the territory with the map. The number 2 is MODELED AS the set {0,1}. But the number 2 is not a set. See for example Benacerraf, What Numbers Could Not Be.
    – user4894
    Dec 30, 2018 at 19:42

3 Answers 3


See First-order arithmetic for the syntactical specifications of the formal language dealing with natural numbers.

The basic arithmetical symbols are : 0 (an individual constant denoting the number zero), the unary successor function s(x), and two binary functions : + and x for the numerical operations of sum and product.

The axioms are the first-order version of Peano axioms.

The usual numerical symbols (the numerals) : 1, 2, 3, ... are introduced as abbreviations :

1=s(0), i.e. "1 is the name for the (unique) successor of the number zero";

2=s(1)=s(s(0)), and so on.

Using the axioms, we prove that :

1+2=3 [which abbreviates : s(0)+s(s(0))=s(s(s(0)))].

  • Same questions as I have for the OP. Sure, that's how we model 1 + 2 = 3 using the Peano axioms. But 1 + 2 = 3 tells us something about the world, while simply definining 1 = S0, 2 = S2, and 3 = S2 doesn't. Meaning is lost when we introduce the abstract mathematical model. I believe that's a problem for the OP's question and also for your answer. This seems to be the heart of the philosophical issue. Abstract models have no meaning. And that's significant. What am I missing?
    – user4894
    Dec 31, 2018 at 2:01
  • No, 1+2=3 tells no more about the world than a mere "link" aka relation between the 3 numbers 1,2 and 3. Of course this link is not arbitrary, it is based on modelling many things happening in reality, but it is no more than a simple link. And, the defining 1=S0 ,2=S1 etc captures the very essence of counting one by one. Dec 31, 2018 at 5:16
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    @user4894 - of course, Peano's axioms are an "abstraction"... but an abstraction of the "real" counting process. We name two the number following one; then we nane three the number following two, and so on. The "following" procedure is exactly what is formalized by the successor funtion. Dec 31, 2018 at 10:04
  • This doesn't really address the underlying problem here or either of the explicit questions. What is the point of the answer?
    – user9166
    Jan 1, 2019 at 23:19

I've found two interesting papers about this issue: 1) Gottlob Frege's "On sense and denotation" https://faculty.washington.edu/smcohen/453/FregeDisplay.pdf

2)Alonzo Church Introduction to mathematical logic (in the introductory chapter about "Names") http://www.hist-analytic.com/ChurchIntroductionIntroduction.pdf

They treat exactly what i asked in a very nice way.

Thanks to @John Forkosh who suggested above the "sense-denotation " duality. It's been a long time since i've reflected about this philosophical issue, and it seemed untractable. But know i've cleared it.


Heres my attempt at a simpler answer, without the need for math:

Your problem arises from your interpretation of equality. You are interpreting = to mean "The thing on the left of the sign is equal in all qualities to the thing on the right, while in math a = sign only signifies that the things on both sides of the sign represent the same quantity.

Likewise, in other fields of math, the equal-sign only compares specific qualities of the things representated by the notation (e.g. that two sets contain the same elements).

  • The vast majority of modern math has nothing to do with quantity
    – user9166
    Jan 3, 2019 at 14:58
  • @jobermark If you give me an example, I'll be happy to delete my answer. Jan 4, 2019 at 7:53
  • The whole of set theory, abstract algebra, topology, symbolic logic, even, perhaps ironically computation theory itself, have to do with what patterns of permutations fit together. Those are not about quantity in the same way the older branches of math like analysis. Since the mid-20th century most of the math done follows from what is left of geometry once you remove measurement and focus on abstract patterns. Since most of math has older roots, it is still about quantities, but most of what is currently being done in math no longer fits into that mold.
    – user9166
    Jan 4, 2019 at 10:54
  • You and the OP are on opposite ends of a continuum where the ansser is in the middle. A lot of what people compute is not numeric. He resents that the focus on computation is even a part of the system. Older views of math focus on numbers as the kinds of things you compute, as a goal, instead of as an incidental part of computing that is focused on some other problem.
    – user9166
    Jan 4, 2019 at 10:59
  • jobemark, do you have math studies? do you currently do math? Or only philosophy? Jan 4, 2019 at 11:16

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