Is it possible to create an axiomatic system where 1+1 doesn't equal 2? What would be the consequences of such a system? [closed]

1+1=2 is a result (perhaps arguably more of a definition than a theorem?) of Peano Arithmetic, as well as other systems such as ZFC. I understand that 1+1 doesn't necessarily have to equal 2 if we consider addition modulo n, for some n in the positive integers. However, my question mainly has to do with whether or not one can create a system in which 1+1 doesn't equal 2 in base 10. Would we have to give the number 10 a different value? Or are "values" of numbers inherent properties of those numbers in all systems? It seems to me like it should be possible to create a system with any axioms I want, as long as they are consistent with each other.

Could such a system derive contradictions? It is possible that it would, but is it possible to create axioms that are actually consistent with one another?

I apologize if any of what I wrote includes misconceptions or things that don't make much sense. Please correct me on anything.

• The question is vacuous without extra conditions. One can trivially come up with a system that has symbols "1", "+" and "2" where 1+1 is not 2. Even ordinary arithmetic will do if we simply relabel 2 and 3 into each other. The base is altogether irrelevant, whether 1+1 is 2 or not has nothing to do with which base is used to generate number symbols, only definitions and axioms relating them matter. Aug 19 '20 at 7:10
• @Conifold Thank you for the clarification. So, if we consider the symbols "1", "+", and "2" to mean what they usually mean when one writes "1+1=2", it is then logically inconsistent to create an axiomatic system where 1+1 doesn't equal 2, right? Because that would contradict the definitions of the symbols. Aug 19 '20 at 7:32
• "What they usually mean" is implicitly defined by the axioms and definitions. In Peano arithmetic "2" is simply an abbreviation for 1+1, so they are equal by definition, and it is inconsistent for it to be otherwise. To make it possible you have to define "2" some other way or introduce it as a basic symbol and have axioms which imply that 1+1 is not 2. But then "2" will not be what it "usually means". Aug 19 '20 at 7:41
• @Conifold Ah, that makes sense. I have a lot to ponder but thank you for the clear explanations. Aug 19 '20 at 7:45
• do you mean an axiomatic system where addition is an invalid operation? Aug 19 '20 at 18:27

4 Answers

Your question is grammatically sound but it sort of defeats itself by suggesting to rewrite definitions but then judge them by the old definition. I'm not accusing you of any illegal substance abuse, but this is what I call a "stoner question". My favorite real-life stoner question that I've been asked is "what if clocks were like ants?"

In both your question and my example, it makes grammatical sense but if you make an active effort to understand the meaning of the question, it becomes clear that the question relies on some presupposed definitions and some radically changed definitions, without a clear indication as to which is which.

To summarize the longer answers below: If you were to concretely express how you are making `1 + 1` to equal `3`, it would no longer be a stoner question but you would have also provided your own answer.

The amount of elaboration necessary to make this question answerable inherently answers the question itself.

There are different ways to redefine `1 + 1 = 2`. Each of them have far reaching consequences, and each set of consequences will be different.

1
You can redefine the `+` operator. For example, we can define this as string concatenation, whereby `1 + 1 = 11`. This doesn't change the mathematical value, it changes what you can do with a mathematical value.

More interestingly, triangular numbers. If `T(x)` is defined as the x-th triangle number, then `T(1+1) = T(2) = 3`. If you redefine `+` to mean "take the triangle number of the sum of the two numbers", then `1 + 1 = 3`.

2
You can redefine how numbers relate to one another. The integer numbers we use are a linear progression, but they could for example be expressed logarithmically or exponentially.
A precedent for this is decibels, which is on a logarithmic scale. For a given amount of decibels `x`, the meaning of `x+1` is that the sound is twice as loud. It didn't increase by a unit of `1` decibel, it doubled.

Similarly, if you put integers on a logarithmic scale, then `x + x` (linear) is equal to `x + 1` (logarithmic).

3
You can change the meaning of the symbols and express this amount of hasthags: ## using the symbol `3`. But that is really just a cop out and not in the spirit of this question, I suspect.

There are more ways to redefine it, but it always boils down to changing the meaning of at least some of the components that make up `1 + 1 = 2`. Subsequently, you have to re-evaluate every mathematical concept that depends on that component you redefined.

Your question is asking what would need to be re-evaluated, but that cannot be answered unless I know which component you've redefined and how you've redefined it.

Would we have to give the number 10 a different value?

When you redefine `1 + 1 = 3`, you have messed with the meaning of numbers and how they relate to one another. That effect isn't limited to the numbers `1`, `2` and `3`. That effect resonates througout the numberline.

Depending on how you redefined it (see above), this can have different consequences. Therefore, I cannot tell you what 10 (the digit sequence) expresses or how to write ten (mathematical amount) in symbols, unless you first define how you changed `1 + 1` to now equal `3`

Or are "values" of numbers inherent properties of those numbers in all systems?

Numbers are really just symbols. But I assume that you are not trying to redefine the symbols themselves, nor the quantities that are expressed by that symbol, but rather how these symbol/quantities relate to one another.

The fact that we use sequences of digits (ranging from 0-9) has no mathematical bearing. As far as mathematics cares, we could have a unique symbol for each number, and the same mathematical truths will remain.

We already understand this, because that's what algebra does. If I tell you that `A + A = B` and that `B + B = C`, then you know for a fact that `A + A + A + A = C`. Which symbol you use to write down these numbers is irrelevant. The only thing that's relevant is the relation between these numbers.

whether or not one can create a system in which 1+1 doesn't equal 2 in base 10.

Every base, when expressed in its own base, is expressed in base "10":

• Binary (base 2), but since (decimal) 2 = (binary) `10`, you can define binary in binary as base `10`
• Octal (base 8), but since (decimal) 8 = (octal) `10`, you can define octal in octal as base `10`
• Hexadecimal (base 16), but since (decimal) 16 = (hexadecimal) `10`, you can define hexadecimal in hexadecimal as base `10`

This repeats for every base `X` you can come up with.

Humans have chosen 10 (the physical amount) as their preferred digit range, but this is a completely arbitrary choice, based on how many individual fingers we have. There is no reason why another civilization wouldn't have picked other numbers. It's a fairly often mentioned fact that South American civilizations chose base 12, leading them to be able to count to 144 (12²) on two hands.

Mathematical truths remain the same, regardless of what base you express them in, or whether you don't express them in a base at all (i.e. a unique representation for every possible value).

It seems to me like it should be possible to create a system with any axioms I want, as long as they are consistent with each other.

I think you are underestimating both the effort required to discover truths from the ground up in your new axiomatic system, and how exponentially difficult it becomes to keep things consistent.

Over the ages, there have been countless mathematical models which were thought to be consistent until it turns out they weren't. The ones that have stood the test of time in some way relate back to `1+1=2`.

If you redefine this axiom, then all bets are off and we start from scratch. Nothing we know about math can be relied on anymore, we have to vet everything from scratch, and figure out how to accurately model this.

• Thank you for the wonderful answer. I just want to briefly summarize my understanding of what you said: 1+1 doesn't have to equal 2 if we redefine the relations between the symbols "1" & "2" and the operation "+". If we were to redefine said symbols, we would change the values of all numbers on the number line (or perhaps only some, depending on how we define the symbols and their relations). Hence, the only thing that matters is how we relate the symbols in regards to their given values, which then forms the basis for the truths of the system. Does this sound right? Aug 19 '20 at 20:16

Classical arithmetics are based on multiple assumptions regarding human perception. 1+1 is not necessarily 2 in other contexts. A simple example: 1+1=2 only if you assume that the objects you are adding are isolated systems.

When you make the operation 1+1 and the systems interact, results different than classical addition can be expected. 1+1 points, added in a space where they can interact are much more than two objects: they are two points+one line, for example. Two persons can become three in nine months. Two H atoms + one O atom become three atoms+one molecule with multiple emergent properties. Ten oranges can be ten oranges or ten oranges in a pyramid. 1+1 bubbles surfaces can become a single one. Two electric potentials added in series produce a different potential to what they would produce if added in parallel.

Classical addition always assume that objects don't change, which is just an ideal generated by reason. But in nature, there's absolutely no thing that remains static. The river is never the same. The person you've looked two seconds ago has already changed. Even rocks mutate continuously. Only in our minds things don't change. So, addition in a natural framework does not follow the same rules as mathematical addition.

Take numbers [0, 1, 2, 3, 4]

Let's define addition as jumping over 1 number: 0+1=2, 2+1=4, 4+1=1, 1+1=3, 3+1=0.

Is this different from just relabelling numbers in ordinary modulo addition? Not sure. Maybe, if we insist that 3 > 2. Is it consistent? Check for yourself.

• You can define an isomorphism from your numbers into the integers modulo 5 by having “4”<-> 0, “3”<-> 1, “2” <-> 2, “1”<-> 3 and “0”<-> 4 Aug 22 '20 at 18:43

To ask about the physical "meaning" of 1+1=2, as Rodolfo suggests, gives you some whimsical options in what we might call "applied math."

For instance, the equal sign = rarely comes in for much attention, but it is a suspicious relation of nonidentity. That particular symbol was created by Robert Recorde in 1557 using two parallel lines because "no two things can be more equal."

Now as it happens, the concept of parallel lines, the fifth postulate, was always considered the most problematic of Euclid's postulates, and its rejection by Gauss and others was the key to modern non-Euclidian geometries.

Like the parallel postulate, the whole idea of a frictionless, reversible relationship symbolized by = has no physical meaning, as Rodolfo notes in his Hericlitean reply.

The statement 1+1=2 has meaning only insofar as it is comprehended, which requires some infinitesimal motion and expenditure of energy. To suppose that it is perfectly reversible violates the Second Law of Thermodynamics.

This suggests that, just as "parallel" lines eventually meet in curved spaces so 1+1=2 will eventually "wear out," becoming more and more unequal with accumulating iterations 1+1 = 2 = 4-2 = 3-1 = ..... So the equation, to have "meaning," may need some time value or memory limit added in by way of correction.

Please note that I have almost no idea of what I'm talking about here, and I'm more of an idealist myself, but I was always interested in the implications of mathematical equality (=), so if anyone knows of some interesting readings along this line, please let me know. Perhaps I'll try to frame this as a future question.