I have been thinking about some pretty speculative material that has been hard to wrap my mind around, but I feel like there is more to discover in this train of thought.
I can't remember exactly how the topic came up, but I was thinking about how sometimes we might be inclined to ask, Why does 1 + 3 = 4?
Actually, maybe this came about because I was thinking about integers. They have very curious, particular properties. It is interesting that integers are defined mathematically as rings. They have one operation, +, which creates group structure, and a second operator, *, which I think with integers is a semi-group, maybe (there is an identity element, but there are not inverses for every element).
I guess I was wondering if the properties of the integers are kind of random, in a way. They come up so often in daily life, but when you look at it as a mathematical object, it just comes across as particular. There are more general, symmetrical objects. Even just the fact that you have two operations but they don't have the same properties. It would feel cleaner and simpler to have two operations that both forms groups, for example. But there are many ways to think about why the integers have the particular properties they do.
I think it led me back to this idea I've been thinking about, which is how it's possibly that mathematical truths are "necessarily necessary". To summarise some other thoughts I've had, it's basically about how if you try to imagine a counterfactual, an alternative, hypothetical reality where there are some different laws of math, there seems to be a philosophical paradox.
It is, that if there is a valid, logical consistent, formulable alternative to some mathematical theorem, then that doesn't really mean we've discovered a "different" kind of math, because the truth of math does not depend on circumstances or states of a particular world. So long as one can recognise, logically, that given certain premises, a certain theorem would follow, then both the original and the opposite form of that theorem would both be valid mathematical theorems, in our world, in one and the same. They are both logically sound. They just start with different premises. In other words, there cannot be a hypothetical alternative math, because anything that is, is just normal math anyway. If a math theorem is true, it is true in all possible worlds, because it is a product of inherent reason, intrinsically necessary logical self-consistency. Logic, whatever it is, by its own definition, has to be valid, to be logic at all. You cannot make an illogical logic, otherwise, it is not a logic.
I believe Wittgenstein especially may have spent some time thinking about this, and I think there's an interesting passage possibly in the Tractates where he says that it is not really a coherent idea to consider what a geometric figure might look like, outside of a geometric "plane". That is, all shapes exist within some... parameters which give rise to them. If you plot 2D space with an X and Y coordinate, you can draw myriad geometric figures in there, any one you like. But it is not merely false to consider a geometric figure which does not occur in the geometric plane. It is more than false. It is more like incoherent. It simply doesn't make sense.
Maybe it's something to do with co-creation, semantic, mutual inter-dependence. If the rules of the geometric plane are themselves what constitute geometric figures, maybe, the issue with thinking of a geometric figure that exists outside of a geometric plane, is maybe, going further than "nonsensical", it is not even meaningful because you don't realise that they are actually the same thing.
I am not sure if I stand by it, but it was a new way to think about it that intrigued me.
Instead of merely saying, that the question of if 3 + 1 could equal something other than 4, in some world, is... impossible because it would be impossible to reconcile such a "fact" with the rest of any logically consistent world (i.e., how do you count to 4, given the properties of arithmetic? if 3 + 1 does not equal 4, then what does it equal? A different number? Or nothing?), if you try to flesh out your view of such a world, it likely seems to fall apart immediately. It cannot be reconciled with a logical world without reinterpreting it so heavily that it loses the original meaning you were aiming for. In that case, you end up back to where you started - you have processed, trimmed, finagled, converted a proposition which is not only false - it is not even well-formed, because you had no idea what it even meant, and you had to think extensively about it, and basically change core aspects of its meaning to suddenly not merely make it true, but to make it even interpretable, even coherent enough to ask whether it's true or not. Before then, you haven't really attained that right yet. It's like Schrodinger's paradox where it's as of yet undecidable what the state of a particle is before observation. A logically incoherent, ill-formed assertion, can not even be examined for its truth content before it is determined if it is even a member of the domain of logic itself, some system that we use. Just like the geometric picture, in some conception of what logic necessarily is, the fundamental form it must take to even be logic, we cannot have a logical assertion that is not of the form of a logical assertion. So it's possible that 1 + 3 does not equal 4 fails to even qualify, not merely as true, but rather, as meaningful.
I still spent a lot of time trying to think that over in my head. I was looking at the figure of four people standing near me. I tried to really realise, and apply, the idea that there is no "why" as to the observation that "1 + 3 = 4". My first way of viewing it was just by reason - I adduced that the counterfactual, 1 + 3 != 4, would be too hard to reconcile with a logical world, and so it simply could never be so.
But that is the conventional way of thinking about tautologies and contradictions. It didn't satisfy me, because even if the opposite is not reasonable, that is not enough to explain why something has the form it does, and not some other. I considered that just because something is true in all possible worlds does not imply it is a tautology. Something that is true in all possible worlds may be true by virtue of some kind of extended reasoning process. It may not be simple. It may be highly deductive, moving from one fact to another. And yet, by flawless reasoning, it is shown to always be logically mandated.
But some things are in a way "true in all possible words", but it's not like that. Like the equation "1 = 1".
Conventionally, we consider that this is a) true, and b) it surely could never not be true.
But as I thought about why 1 + 3 = 4... I considered that surely it cannot be that there is some "rule" governing our universe that decreed this order, this relationship. I was thinking about how, if my understanding of the necessary necessity of mathematical truths is to be accepted or assumed, it basically means that whatever the foundations of mathematics or logical truth are, they cannot be propositions that have a conceivable formulable alternative. Like, there is Euclidean geometry, and non-Euclidean geometry. They are two systems of geometry with different properties. They are the result of striking out only one of the axioms - a logical basis, a proposition - from Euclidean geometry. That is what I mean by contingent. The axioms of Euclidean geometry are apparently contingent, because it is mentally possible to form an alternative. As I said at the beginning, the totality of mathematics contains all logical possibilities. Any contingent statement is joined by its contingent alternatives within the bracket of "all of mathematics". Which is why if mathematics has a basis, it cannot be a contingent statement, but must be something that it is not possible to even express an alternative formulation of.
It kind of hints at this weird idea that just like a geometric figure, we should not get confused about the minor question of if a proposition happens to be the case in a particular world - this is a minor, trivial attribute, like an afterthought, to be easily ascertained through observation - but it's almost like what we conventionally call "true" is one and the same thing as "well-formed", because if logic is the domain of "truth" (under a certain interpretation of "truth"), then it can't be logic if it's not "true" (in the same slightly unconventional interpretation of that word).
So as I looked at those 4 people, I was thinking that I cannot think of 1 + 3 = 4 as being assertive at all, as having propositional content at all. If I held this to be a self-evident truth of the world, deeply difficult to justify why, then, according to my thinking process above... it's like we have to look at numbers in a way with the least prior assumptions possible - our culture, words, symbols. If we look at some form itself, like a human, its kind of like we can imagine without even the concept of "numbers", that there is a form, and then a form again, and then a form again, and then a form again. In this way of looking... in a world almost without language, there is no question "why" to even be asked. If someone said "why", we could say, "why what?" In a way, by observing some form, some additional presence of some form, I think we observe what I was trying to say, that it could never be the case that 1 + 3 could not equal 4, because if you study where those concepts even came from (the observation of recurrence), it is intrinsically necessary that after each one comes another, and there you have it. If you want to give names to them, sure, there's 1, there's 2, then 3, then 4. But this was not a phenomenon that required an explanation, to occur. It created its own existence, in a way.
So, it's almost more like humans are under an illusion of language, just because they can write an equation "1 + 3 = 4" - it makes them think that 1, 3, 4, +, and = are all highly distinct entities, each with unique identity, bound up in a complex, non-arbitrary, peculiar system of relationship to each other.
But in that non-verbal way of looking at something, it's the opposite. As I said above, it is almost (kind of a trippy, or semi-Buddhist, thought) as if they are the same thing. It is very much like how Wittgenstein says you can't have a concept of the "King" in chess, without the entire context of chess itself. The meaning of the concept of "4" necessarily derives from the precondition of some previous concepts, 1, 2 and 3. And in a weird way of thinking about it, it makes it easier to see it no longer as contingent, as distinct things requiring a "why" for why they get put together, honestly, to stop thinking of them as separate things. It's almost like the set of all numbers, all numbers together, are one single thing. You couldn't have the very fundamental concept of "numbers" if you left one of the numbers we know out. The mystery of contingency disappears. It would be impossible for 1 + 3 to not equal 4, exactly like the impossibility of a geometric figure outside the parameters of geometry. It isn't simply false. I feel like it needs a stronger word than "necessary" or "tautological" or even "not well formed"... something like, "it is intrinsically impossible given the very definitive criteria of the concept of numbers, themselves". I think I prefer that way of saying it, I think it says something more, than the other epithets.
But this is where this train of thought led me.
When you write a tautological equation in mathematics, like 1 = 1, one really interesting thing about that is that you can algebraically manipulate its form in any way you like. Subtract one from both sides, now you have 0 = 0.
Isn't it strange that apparently, the assertion 1 = 1 is the same statement as 0 = 0? When we normally do algebra, we use algebraic operations to transform the form of a proposition, but only in order to see it in a clearer form, to assess the truth of it. It is not the case that by doing algebraic operations on both sides of some conjecture, that the mathematics drips into a puddle like melting ice cream, that we can send the form in any way we want, almost like we can collapse any inherent information in the proposition by validly transforming it into any other mathematical expression we want.
This is very interesting to me. It adds to what I was saying above, about the idea that "1=1" being a tautology because it is true in all possible worlds, as possibly being deeply limiting and misleading. Actually, "1 = 1" is not meaningful. It hasn't entered the "form" of a proposition, because a proposition is (somehow, in some way I haven't thought about yet), something that has inherent differentiability. There is some intrinsically logical, unavoidable way in which if a proposition is true, it changes some aspect or characteristic of a world; some way in which each valid proposition is distinct, not indistinguishable, from all other propositions.
None of this is true for 1 = 1. We showed that somehow, 1 = 1 can be validly manipulated into any expression we want, like 3 + 4 = 3 + 4. Also, it cannot possibly change the state of information in any way. Maybe I am making a mistake here, but I feel like there could be a difference between that, vs. a tautology that is actually still meaningful - somehow, an assertion that is necessarily true, but it isn't deceptively cheating through a meaninglessly senseless form, it still says something about the world.
This is where my thought led me to.
While there are many systems of formal logic with various rules and axioms, has anyone ever developed a formal logic in which certain kinds of tautologies are not considered well-formed? Some theory of logic in which "1 = 1" is not considered part of the domain of logic, is not a logical expression, because it is actually meaningless (in the way I tried to show above), because it does not actually meet a key, fundamental criteria of the forms of true statements themselves?