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.
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.
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).
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
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
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
- Octal (base 8), but since (decimal) 8 = (octal)
10, you can define octal in octal as base
- Hexadecimal (base 16), but since (decimal) 16 = (hexadecimal)
10, you can define hexadecimal in hexadecimal as base
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
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.