Troglodite thinking:

Opinion - Something you believe to be true based on no or limited evidence.

Fact - A thing you believe to be true that is objectively true.

Enlightened thinking:

Opinion: Anything I believe to be true whether it is true or not and no matter what level of evidence I believe I have.

Fact - A thing which an oracle could, would know is objectively true, but since oracles dont exist we can never exert with certainty that something is or is not fact. It is at best an abstract concept.

@freemo While I agree with this in general, I feel like there's got to be a limit. Is "1+1=2" opinion?

@LouisIngenthron

Are we talking purely mathematically, in which case it is only true by definition, and it is still your opinion in how you are understanding that definition. For example in computing (not pure math) we have a saying "2+2 = 5 for sufficiently large values of 2" which is a play on some of the particulars of computing.

But something being true by definition is only true due to circular logic, and therefore true but not int he sense of being a world fact (a true idea).

By contrast though if you mean the real world manifistation of that.. so "if i have one rock, and pick up another, then I have two rocks" we still wind up in the same problem of true by definition but its just a harder concept to understand. In this case what is a "rock" and how we count rocks are ultimately at play and it still becomes a situation where it is only true by virtue of the fact that we define it as such. Again we would be in the problem that the definition still has room for definition... if i pick up a pile of sand, how many rocks do I have? One could argue each grain of sand counts as a rock.

@freemo Right, but in that example, it's "rock" that becomes ambiguous, not "1", which is an abstract concept we can reliably apply to rocks, apples, bananas, whatever.

I believe that somewhere between the theory and the individual applications, there lies an objective truth that remains factual regardless of our perceptions or ability to observe it.

@LouisIngenthron also to address your otherpoint.. my argument is not that there isnt an objective truth, only that no oracle exists as to what is and is not an objective truth. So even if something is, in reality an objective truth, the fact that it is so is still just your opinion of it.

@freemo Idk about that. If you met someone from 100 years ago, you'd sure seem like an oracle to them. I imagine that's largely true going back through human history. As a species, we seek to define and understand the world around us, and the only way we can truly do that is by finding these universal truths and using them as lenses. I don't think we've found many yet, but we're working hard on finding more.

I don't need confirmation from an omniscient oracle to confirm a fact as true. I just need to be reasonably certain that even totally foreign beings who experience life in a way that's unfathomable to me would still inevitably and independently come to the same conclusion, and I believe that to be true of "1+1=2".

@LouisIngenthron Also it bears mentioning that as much as you may believe 1+1=2 is a universally and objectivelly true fact it is not. It is only true under an explicit definition of that being so, and depends on every definition therewithin.

For example 1+1 does nto equal two in the following systems:

in base 2 systems there is no symbol such as 2 so "1+1=2" is patently false. however "1+1=10" is true under that system. This is because our definition of the numbers and how they expressed is different.

In braurer groups addition is defined as a tensor product over algebras. Without getting too technical under this mathematical ring "1+1 does not equal 2" in fact this very assertion is non nonsensical.

In noncommunicative addition rings then "1+2 is not equal to 2+1" and in many such rings 1+1 equals something other than 2 as a consequence (in some such rings it does).

Similarly in mathemetical rings the very set of numbers that exist may be finite, and the number 2 may not exist at all. In fact you can have mathematical rings where 0 and 1 are your only numbers, and as such "1+1=2" is not true.

@freemo I feel like you're unnecessarily equating labels with their underlying concepts here.

The concept of two is still two whether we write it as "2" in base-10 or as "10" in base-2.

Likewise, the concept of integer addition is distinct from the addition symbol "+" which is also used to denote many similar-yet-distinct concepts (such as the ones you describe). While those may be referred to as "addition", they aren't the concept of integer addition that I'm specifically referring to.

These are, as you initially described them, effectively "definitions" that prove themselves circularly, but a tiny subset of such definitions describe concepts that would seem to be universally constant, that are independently and repeatably verifiable regardless of perspective. If we don't call those "facts", what do we call them?

@LouisIngenthron

> I feel like you're unnecessarily equating labels with their underlying concepts here.

> The concept of two is still two whether we write it as "2" in base-10 or as "10" in base-2.

No this is exactly what i said at the offset, there are two ways to discuss this, both very different but both agree with what I'm saying here.

What i just expressed was the "by mathemical definition" where I showed "1+1=2" is **not** a universal truth, it is only true when defined to be true, and not in any sense in reality.

> Likewise, the concept of integer addition is distinct from the addition symbol "+" which is also used to denote many similar-yet-distinct concepts (such as the ones you describe). While those may be referred to as "addition", they aren't the concept of integer addition that I'm specifically referring to.

So lets use the other half of the coin, since that is what you are asserting here you mean. Not by definition but due toi the real world concepts they represent.

> These are, as you initially described them, effectively "definitions" that prove themselves circularly, but a tiny subset of such definitions describe concepts that would seem to be universally constant, that are independently and repeatably verifiable regardless of perspective. If we don't call those "facts", what do we call them?

So even in the real world, not by definition, the real world concept that "1+1=2" is not a universal fact in reality either.

I mean sure, if I have one duck, and add to that one apple, I now have two things... but again that is only because we define what a duck is, and where one thing ends and another starts, its still all by linguistic definition, and ONLY works for some things even if we accept their definittions...

Here are all the counter examples where "1+1 does not equal 0":

One electron added to one positron results in 0 physical things. So in this scenario "1+1=0"

one blob of water added to another blob of water results in a single blob of water, therefore "1+1=1"

If you put two humans who are attractive to eachother in a room and wait 9 months you get an extra human. Therefore in some cases "1+1=3"

@freemo In none of those examples are you doing integer addition.
The first two examples are collision, not addition. The third is procreation, not addition.

One electron added to one positron is two physical things... until they collide, which kicks off annihilation (which is also not addition).

For water, your quantity is not "1", but rather "blob". So, yes, blob+blob=blob, but none of those are integers. When they combine, it's not the math that changes, but rather the means by which you measure it. In different units, such as one gallon plus one gallon, the math holds true.

These universal truths I describe are often incredibly narrowly defined, by necessity. I believe very few things are truly absolute, but there are a few.